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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 38707 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs )) | |
2 | cnvepresex 38313 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | |
3 | eldisjsdisj 38706 | . . . 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 38686 | . 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 2108 Vcvv 3479 E cep 5581 ◡ccnv 5682 ↾ cres 5685 Disjs cdisjs 38193 Disj wdisjALTV 38194 ElDisjs celdisjs 38195 ElDisj weldisj 38196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 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-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 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 38390 df-rels 38464 df-ssr 38477 df-cnvrefs 38504 df-cnvrefrels 38505 df-cnvrefrel 38506 df-disjss 38682 df-disjs 38683 df-disjALTV 38684 df-eldisjs 38685 df-eldisj 38686 |
This theorem is referenced by: (None) |
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