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| 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 4895 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
| 2 | 1 | necon1ai 2967 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| 3 | dfopg 4870 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
| 4 | snex 5435 | . . . . 5 ⊢ {𝐴} ∈ V | |
| 5 | 4 | prnz 4776 | . . . 4 ⊢ {{𝐴}, {𝐴, 𝐵}} ≠ ∅ | 
| 6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ≠ ∅) | 
| 7 | 3, 6 | eqnetrd 3007 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ≠ ∅) | 
| 8 | 2, 7 | impbii 209 | 1 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∅c0 4332 {csn 4625 {cpr 4627 〈cop 4631 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 | 
| This theorem is referenced by: opnzi 5478 opeqex 5502 opelopabsb 5534 setsfun0 17210 fmlaomn0 35396 | 
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