The remarkable reasoning and code generation capabilities of large language models (LLMs) have recently motivated increasing interest in automating formal verification (FV), a process that ensures hardware correctness through mathematically precise assertions but remains highly labor-intensive, particularly through the translation of natural language into SystemVerilog Assertions (NL-to-SVA). However, LLMs still struggle with SVA generation due to limited training data and the intrinsic complexity of FV operators. Consequently, a more efficient and robust methodology for ensuring correct SVA operator selection is essential for producing functionally correct assertions. To address these challenges, we introduce FVRuleLearner, an Operator-Level Rule (Op-Rule) learning framework built on a novel Operator Reasoning Tree (OP-Tree), which models SVA generation as structured, interpretable reasoning. FVRuleLearner operates in two complementary phases: (1) Training: it constructs OP-Tree that decomposes NL-to-SVA alignment into fine-grained, operator-aware questions, combining reasoning paths that lead to correct assertions; and (2) Testing: it performs operator-aligned retrieval to fetch relevant reasoning traces from the learned OP-Tree and generate new rules for unseen specifications. In the comprehensive studies, the proposed FVRuleLearner outperforms the state-of-the-art baseline by 3.95% in syntax correctness and by 31.17% in functional correctness on average. Moreover, FVRuleLearner successfully reduces an average of 70.33% of SVA functional failures across diverse operator categories through a functional taxonomy analysis, showing the effectiveness of applying learned OP-Tree to the Op-Rule generations for unseen NL-to-SVA tasks. These results establish FVRuleLearner as a new paradigm for domain-specific reasoning and rule learning in formal verification.