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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelid | Structured version Visualization version GIF version |
Description: The identity relation is an equivalence relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
eqvrelid | ⊢ EqvRel I |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjALTVid 37613 | . . 3 ⊢ Disj I | |
2 | 1 | disjimi 37640 | . 2 ⊢ EqvRel ≀ I |
3 | cossid 37338 | . . 3 ⊢ ≀ I = I | |
4 | 3 | eqvreleqi 37461 | . 2 ⊢ ( EqvRel ≀ I ↔ EqvRel I ) |
5 | 2, 4 | mpbi 229 | 1 ⊢ EqvRel I |
Colors of variables: wff setvar class |
Syntax hints: I cid 5572 ≀ ccoss 37031 EqvRel weqvrel 37048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-coss 37269 df-refrel 37370 df-cnvrefrel 37385 df-symrel 37402 df-trrel 37432 df-eqvrel 37443 df-disjALTV 37563 |
This theorem is referenced by: (None) |
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