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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opelid | Structured version Visualization version GIF version |
Description: Characterization of the ordered pair elements of the identity relation when the intersection of their components are sets. Note that the antecedent is more general than either component being a set. (Contributed by BJ, 29-Mar-2020.) |
Ref | Expression |
---|---|
bj-opelid | ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-inexeqex 34473 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | ex 415 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))) |
3 | bj-opelidb 34471 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) | |
4 | simpr 487 | . . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
5 | ancr 549 | . . . 4 ⊢ ((𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝐴 = 𝐵 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))) | |
6 | 4, 5 | impbid2 228 | . . 3 ⊢ ((𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)) |
7 | 3, 6 | syl5bb 285 | . 2 ⊢ ((𝐴 = 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) |
8 | 2, 7 | syl 17 | 1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ I ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3493 ∩ cin 3932 〈cop 4570 I cid 5456 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3495 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-opab 5126 df-id 5457 |
This theorem is referenced by: bj-ideqg 34476 bj-opelidres 34480 bj-opelidb1ALT 34485 bj-elid4 34487 |
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