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| Mirrors > Home > MPE Home > Th. List > opnzi | Structured version Visualization version GIF version | ||
| Description: An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
| Ref | Expression |
|---|---|
| opth1.1 | ⊢ 𝐴 ∈ V |
| opth1.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| opnzi | ⊢ 〈𝐴, 𝐵〉 ≠ ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opth1.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | opth1.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | opnz 5478 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
| 4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ 〈𝐴, 𝐵〉 ≠ ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 〈cop 4632 |
| 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 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 |
| This theorem is referenced by: opelopabsb 5535 0nelopab 5572 0nelxp 5719 unixp0 6303 funopsn 7168 cnfldfun 21378 cnfldfunOLD 21391 fmlaomn0 35395 finxpreclem2 37391 finxp0 37392 finxpreclem6 37397 |
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