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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 3484 | . 2 ⊢ 𝑥 ∈ V | |
| 2 | bj-opelidb1 37154 | . 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 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 I cid 5577 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-id 5578 |
| This theorem is referenced by: (None) |
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