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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ideqg | Structured version Visualization version GIF version | ||
| Description: Characterization of the
classes related by the identity relation when
     their intersection is a set.  Note that the antecedent is more general
     than either class being a set.  (Contributed by NM, 30-Apr-2004.)  Weaken
     the antecedent to sethood of the intersection.  (Revised by BJ,
     24-Dec-2023.) TODO: replace ideqg 5861, or at least prove ideqg 5861 from it. | 
| Ref | Expression | 
|---|---|
| bj-ideqg | ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-br 5143 | . 2 ⊢ (𝐴 I 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ I ) | |
| 2 | bj-opelid 37158 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) | |
| 3 | 1, 2 | bitrid 283 | 1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 〈cop 4631 class class class wbr 5142 I cid 5576 | 
| 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-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 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 | 
| This theorem is referenced by: bj-ideqb 37161 | 
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