Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > opnzi | Unicode version |
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4209). (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opth1.1 | |
opth1.2 |
Ref | Expression |
---|---|
opnzi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth1.1 | . . 3 | |
2 | opth1.2 | . . 3 | |
3 | opm 4209 | . . 3 | |
4 | 1, 2, 3 | mpbir2an 931 | . 2 |
5 | n0r 3420 | . 2 | |
6 | 4, 5 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wex 1479 wcel 2135 wne 2334 cvv 2724 c0 3407 cop 3576 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-v 2726 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 |
This theorem is referenced by: 0nelxp 4629 0neqopab 5881 |
Copyright terms: Public domain | W3C validator |