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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 4901 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
2 | 1 | necon1ai 2966 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | dfopg 4876 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | snex 5442 | . . . . 5 ⊢ {𝐴} ∈ V | |
5 | 4 | prnz 4782 | . . . 4 ⊢ {{𝐴}, {𝐴, 𝐵}} ≠ ∅ |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ≠ ∅) |
7 | 3, 6 | eqnetrd 3006 | . 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 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 {csn 4631 {cpr 4633 〈cop 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: opnzi 5485 opeqex 5508 opelopabsb 5540 setsfun0 17206 fmlaomn0 35375 |
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