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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelsymrel | Structured version Visualization version GIF version |
Description: An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.) |
Ref | Expression |
---|---|
eqvrelsymrel | ⊢ ( EqvRel 𝑅 → SymRel 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eqvrel 35853 | . 2 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | |
2 | 1 | simp2bi 1141 | 1 ⊢ ( EqvRel 𝑅 → SymRel 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 RefRel wrefrel 35492 SymRel wsymrel 35498 TrRel wtrrel 35501 EqvRel weqvrel 35503 |
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 209 df-an 399 df-3an 1084 df-eqvrel 35853 |
This theorem is referenced by: eqvrelim 35869 eqvrelsym 35873 |
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