![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4230). (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 4230 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | 1, 2, 3 | mpbir2an 942 | . 2 ⊢ ∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 |
5 | n0r 3436 | . 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 1492 ∈ wcel 2148 ≠ wne 2347 Vcvv 2737 ∅c0 3422 〈cop 3594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 |
This theorem is referenced by: 0nelxp 4650 0neqopab 5913 |
Copyright terms: Public domain | W3C validator |