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Mirrors > Home > ILE Home > Th. List > opnzi | GIF version |
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4156). (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 | . . 3 ⊢ 𝐴 ∈ V | |
2 | opth1.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | opm 4156 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 926 | . 2 ⊢ ∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 |
5 | n0r 3376 | . 2 ⊢ (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 → 〈𝐴, 𝐵〉 ≠ ∅) | |
6 | 4, 5 | ax-mp 5 | 1 ⊢ 〈𝐴, 𝐵〉 ≠ ∅ |
Colors of variables: wff set class |
Syntax hints: ∃wex 1468 ∈ wcel 1480 ≠ wne 2308 Vcvv 2686 ∅c0 3363 〈cop 3530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: 0nelxp 4567 0neqopab 5816 |
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