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| Mirrors > Home > MPE Home > Th. List > Mathboxes > disjALTVid | Structured version Visualization version GIF version | ||
| Description: The class of identity relations is disjoint. (Contributed by Peter Mazsa, 20-Jun-2021.) | 
| Ref | Expression | 
|---|---|
| disjALTVid | ⊢ Disj I | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cosscnvid 38483 | . . 3 ⊢ ≀ ◡ I = I | |
| 2 | 1 | eqimssi 4043 | . 2 ⊢ ≀ ◡ I ⊆ I | 
| 3 | reli 5835 | . 2 ⊢ Rel I | |
| 4 | dfdisjALTV2 38716 | . 2 ⊢ ( Disj I ↔ ( ≀ ◡ I ⊆ I ∧ Rel I )) | |
| 5 | 2, 3, 4 | mpbir2an 711 | 1 ⊢ Disj I | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3950 I cid 5576 ◡ccnv 5683 Rel wrel 5689 ≀ ccoss 38183 Disj wdisjALTV 38217 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-coss 38413 df-cnvrefrel 38529 df-disjALTV 38707 | 
| This theorem is referenced by: disjALTVidres 38758 disjALTVinidres 38759 disjALTVxrnidres 38760 eqvrelid 38791 detid 38795 eqvrelcossid 38796 petid2 38818 | 
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