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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 5367 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ 〈𝐴, 𝐵〉 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∅c0 4293 〈cop 4575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 |
This theorem is referenced by: opelopabsb 5419 0nelopab 5454 0nelxp 5591 unixp0 6136 funopsn 6912 cnfldfunALT 20560 fmlaomn0 32639 finxpreclem2 34673 finxp0 34674 finxpreclem6 34679 |
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