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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 5330 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 710 | 1 ⊢ 〈𝐴, 𝐵〉 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 〈cop 4531 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 |
This theorem is referenced by: opelopabsb 5382 0nelopab 5417 0nelxp 5553 unixp0 6102 funopsn 6887 cnfldfunALT 20104 fmlaomn0 32750 finxpreclem2 34807 finxp0 34808 finxpreclem6 34813 |
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