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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opelidb1 | Structured version Visualization version GIF version |
Description: Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 34468 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.) |
Ref | Expression |
---|---|
bj-opelidb1 | ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an32 644 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V)) | |
2 | bj-opelidb 34468 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) | |
3 | eleq1 2899 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
4 | 3 | biimpac 481 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) |
5 | 4 | pm4.71i 562 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V)) |
6 | 1, 2, 5 | 3bitr4i 305 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 〈cop 4566 I cid 5452 |
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 5196 ax-nul 5203 ax-pr 5323 |
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 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 df-id 5453 |
This theorem is referenced by: bj-idres 34476 bj-elid3 34483 bj-eldiag2 34493 |
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