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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 4868 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 957 | 1 ⊢ 〈𝐴, 𝐵〉 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: opelopabsb 4910 0nelxp 5067 0nelxpOLD 5068 unixp0 5586 funopsn 6319 0neqopab 6596 finxpreclem2 32403 finxp0 32404 finxpreclem6 32409 |
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