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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 4061). (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opth1.1 |
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opth1.2 |
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Ref | Expression |
---|---|
opnzi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth1.1 |
. . 3
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2 | opth1.2 |
. . 3
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3 | opm 4061 |
. . 3
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4 | 1, 2, 3 | mpbir2an 888 |
. 2
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5 | n0r 3296 |
. 2
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6 | 4, 5 | ax-mp 7 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 |
This theorem is referenced by: 0nelxp 4465 0neqopab 5694 |
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