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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjeq | Structured version Visualization version GIF version |
Description: Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021.) |
Ref | Expression |
---|---|
disjeq | ⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 4036 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | 1 | disjssd 38332 | . 2 ⊢ (𝐴 = 𝐵 → ( Disj 𝐴 → Disj 𝐵)) |
3 | eqimss 4035 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
4 | 3 | disjssd 38332 | . 2 ⊢ (𝐴 = 𝐵 → ( Disj 𝐵 → Disj 𝐴)) |
5 | 2, 4 | impbid 211 | 1 ⊢ (𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 Disj wdisjALTV 37810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-coss 38010 df-cnvrefrel 38126 df-funALTV 38281 df-disjALTV 38304 |
This theorem is referenced by: disjeqi 38334 disjeqd 38335 disjdmqseqeq1 38336 |
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