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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 35623 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 35623 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵)) | |
3 | eleq1 2825 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
4 | 3 | biimpac 479 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) |
5 | 4 | pm4.71i 560 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ V ∧ 𝐴 = 𝐵) ∧ 𝐵 ∈ V)) |
6 | 1, 2, 5 | 3bitr4i 302 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 〈cop 4592 I cid 5530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-opab 5168 df-id 5531 |
This theorem is referenced by: bj-idres 35631 bj-elid3 35638 bj-eldiag2 35648 |
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