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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 4942 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 955 | 1 ⊢ 〈𝐴, 𝐵〉 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: opelopabsb 4985 0nelxp 5143 0nelxpOLD 5144 unixp0 5669 funopsn 6413 0neqopab 6698 cnfldfunALT 19759 finxpreclem2 33227 finxp0 33228 finxpreclem6 33233 |
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