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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 3506 | . . . . 5 | |
2 | 1 | eleq2i 2184 | . . . 4 |
3 | 2 | exbii 1569 | . . 3 |
4 | abid 2105 | . . . 4 | |
5 | 4 | exbii 1569 | . . 3 |
6 | 3, 5 | bitri 183 | . 2 |
7 | 19.42v 1862 | . . 3 | |
8 | df-3an 949 | . . . 4 | |
9 | 8 | exbii 1569 | . . 3 |
10 | df-3an 949 | . . 3 | |
11 | 7, 9, 10 | 3bitr4ri 212 | . 2 |
12 | 3simpa 963 | . . 3 | |
13 | id 19 | . . . 4 | |
14 | snexg 4078 | . . . . . 6 | |
15 | 14 | adantr 274 | . . . . 5 |
16 | prmg 3614 | . . . . 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 947 wex 1453 wcel 1465 cab 2103 cvv 2660 csn 3497 cpr 3498 cop 3500 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 |
This theorem is referenced by: opnzi 4127 opeqex 4141 cnm 7608 setsfun0 11906 |
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