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Mirrors > Home > ILE Home > Th. List > opm | Unicode version |
Description: An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.) |
Ref | Expression |
---|---|
opm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-op 3536 | . . . . 5 | |
2 | 1 | eleq2i 2206 | . . . 4 |
3 | 2 | exbii 1584 | . . 3 |
4 | abid 2127 | . . . 4 | |
5 | 4 | exbii 1584 | . . 3 |
6 | 3, 5 | bitri 183 | . 2 |
7 | 19.42v 1878 | . . 3 | |
8 | df-3an 964 | . . . 4 | |
9 | 8 | exbii 1584 | . . 3 |
10 | df-3an 964 | . . 3 | |
11 | 7, 9, 10 | 3bitr4ri 212 | . 2 |
12 | 3simpa 978 | . . 3 | |
13 | id 19 | . . . 4 | |
14 | snexg 4108 | . . . . . 6 | |
15 | 14 | adantr 274 | . . . . 5 |
16 | prmg 3644 | . . . . 5 | |
17 | 15, 16 | syl 14 | . . . 4 |
18 | 13, 17, 10 | sylanbrc 413 | . . 3 |
19 | 12, 18 | impbii 125 | . 2 |
20 | 6, 11, 19 | 3bitr2i 207 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3a 962 wex 1468 wcel 1480 cab 2125 cvv 2686 csn 3527 cpr 3528 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-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-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: opnzi 4157 opeqex 4171 cnm 7640 setsfun0 11995 |
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