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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelrefrel | Structured version Visualization version GIF version |
Description: An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.) |
Ref | Expression |
---|---|
eqvrelrefrel | ⊢ ( EqvRel 𝑅 → RefRel 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eqvrel 36321 | . 2 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | |
2 | 1 | simp1bi 1146 | 1 ⊢ ( EqvRel 𝑅 → RefRel 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 RefRel wrefrel 35962 SymRel wsymrel 35968 TrRel wtrrel 35971 EqvRel weqvrel 35973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1090 df-eqvrel 36321 |
This theorem is referenced by: (None) |
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