Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4362 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
2 | 1 | necon1ai 2809 | . 2 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | dfopg 4338 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | snex 4835 | . . . . 5 ⊢ {𝐴} ∈ V | |
5 | 4 | prnz 4253 | . . . 4 ⊢ {{𝐴}, {𝐴, 𝐵}} ≠ ∅ |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ≠ ∅) |
7 | 3, 6 | eqnetrd 2849 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ≠ ∅) |
8 | 2, 7 | impbii 198 | 1 ⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 {csn 4125 {cpr 4127 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: opnzi 4869 opeqex 4887 opelopabsb 4910 setsfun0 15726 |
Copyright terms: Public domain | W3C validator |