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Mirrors > Home > MPE Home > Th. List > opnz | Structured version Visualization version GIF version |
Description: An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opnz | ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprc 4892 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅) | |
2 | 1 | necon1ai 2964 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | dfopg 4867 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}) | |
4 | snex 5427 | . . . . 5 ⊢ {𝐴} ∈ V | |
5 | 4 | prnz 4777 | . . . 4 ⊢ {{𝐴}, {𝐴, 𝐵}} ≠ ∅ |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ≠ ∅) |
7 | 3, 6 | eqnetrd 3004 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ≠ ∅) |
8 | 2, 7 | impbii 208 | 1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ≠ wne 2936 Vcvv 3470 ∅c0 4318 {csn 4624 {cpr 4626 ⟨cop 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 |
This theorem is referenced by: opnzi 5470 opeqex 5494 opelopabsb 5526 setsfun0 17134 fmlaomn0 34994 |
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