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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ideqb | Structured version Visualization version GIF version |
Description: Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.) |
Ref | Expression |
---|---|
bj-ideqb | ⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5839 | . . 3 ⊢ Rel I | |
2 | 1 | brrelex1i 5745 | . 2 ⊢ (𝐴 I 𝐵 → 𝐴 ∈ V) |
3 | inex1g 5325 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V) | |
4 | bj-ideqg 37140 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ V → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
6 | 2, 5 | biadanii 822 | 1 ⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 class class class wbr 5148 I cid 5582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 |
This theorem is referenced by: (None) |
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