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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eleldisjseldisj | Structured version Visualization version GIF version |
Description: The element of the disjoint elements class and the disjoint elementhood predicate are the same, that is (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴) when 𝐴 is a set. (Contributed by Peter Mazsa, 23-Jul-2023.) |
Ref | Expression |
---|---|
eleldisjseldisj | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleldisjs 38671 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs )) | |
2 | cnvepresex 38277 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | |
3 | eldisjsdisj 38670 | . . . 4 ⊢ ((◡ E ↾ 𝐴) ∈ V → ((◡ E ↾ 𝐴) ∈ Disjs ↔ Disj (◡ E ↾ 𝐴))) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) ∈ Disjs ↔ Disj (◡ E ↾ 𝐴))) |
5 | 1, 4 | bitrd 279 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ Disj (◡ E ↾ 𝐴))) |
6 | df-eldisj 38650 | . 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | |
7 | 5, 6 | bitr4di 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ ElDisj 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2104 Vcvv 3477 E cep 5581 ◡ccnv 5682 ↾ cres 5685 Disjs cdisjs 38155 Disj wdisjALTV 38156 ElDisjs celdisjs 38157 ElDisj weldisj 38158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-clab 2711 df-cleq 2725 df-clel 2812 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-eprel 5582 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-coss 38354 df-rels 38428 df-ssr 38441 df-cnvrefs 38468 df-cnvrefrels 38469 df-cnvrefrel 38470 df-disjss 38646 df-disjs 38647 df-disjALTV 38648 df-eldisjs 38649 df-eldisj 38650 |
This theorem is referenced by: (None) |
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