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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elid3 | Structured version Visualization version GIF version |
Description: Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.) |
Ref | Expression |
---|---|
bj-elid3 | ⊢ (〈𝑥, 𝐴〉 ∈ I ↔ 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3481 | . 2 ⊢ 𝑥 ∈ V | |
2 | bj-opelidb1 37135 | . 2 ⊢ (〈𝑥, 𝐴〉 ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝐴)) | |
3 | 1, 2 | mpbiran 709 | 1 ⊢ (〈𝑥, 𝐴〉 ∈ I ↔ 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 Vcvv 3477 〈cop 4636 I cid 5581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5210 df-id 5582 |
This theorem is referenced by: (None) |
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