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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 3407 | . . . . 5 | |
2 | 1 | eleq2i 2145 | . . . 4 |
3 | 2 | exbii 1536 | . . 3 |
4 | abid 2069 | . . . 4 | |
5 | 4 | exbii 1536 | . . 3 |
6 | 3, 5 | bitri 182 | . 2 |
7 | 19.42v 1827 | . . 3 | |
8 | df-3an 921 | . . . 4 | |
9 | 8 | exbii 1536 | . . 3 |
10 | df-3an 921 | . . 3 | |
11 | 7, 9, 10 | 3bitr4ri 211 | . 2 |
12 | 3simpa 935 | . . 3 | |
13 | id 19 | . . . 4 | |
14 | snexg 3956 | . . . . . 6 | |
15 | 14 | adantr 270 | . . . . 5 |
16 | prmg 3511 | . . . . 5 | |
17 | 15, 16 | syl 14 | . . . 4 |
18 | 13, 17, 10 | sylanbrc 408 | . . 3 |
19 | 12, 18 | impbii 124 | . 2 |
20 | 6, 11, 19 | 3bitr2i 206 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 w3a 919 wex 1421 wcel 1433 cab 2067 cvv 2601 csn 3398 cpr 3399 cop 3401 |
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-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 |
This theorem is referenced by: opnzi 3990 opeqex 4004 |
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