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Mirrors > Home > ILE Home > Th. List > opth1 | Unicode version |
Description: Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opth1.1 | |
opth1.2 |
Ref | Expression |
---|---|
opth1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth1.1 | . . . 4 | |
2 | 1 | sneqr 3552 | . . 3 |
3 | 2 | a1i 9 | . 2 |
4 | opth1.2 | . . . . . . . . 9 | |
5 | 1, 4 | opi1 3987 | . . . . . . . 8 |
6 | id 19 | . . . . . . . 8 | |
7 | 5, 6 | syl5eleq 2167 | . . . . . . 7 |
8 | oprcl 3594 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 |
10 | 9 | simpld 110 | . . . . 5 |
11 | prid1g 3496 | . . . . 5 | |
12 | 10, 11 | syl 14 | . . . 4 |
13 | eleq2 2142 | . . . 4 | |
14 | 12, 13 | syl5ibrcom 155 | . . 3 |
15 | elsni 3416 | . . . 4 | |
16 | 15 | eqcomd 2086 | . . 3 |
17 | 14, 16 | syl6 33 | . 2 |
18 | dfopg 3568 | . . . . 5 | |
19 | 7, 8, 18 | 3syl 17 | . . . 4 |
20 | 7, 19 | eleqtrd 2157 | . . 3 |
21 | elpri 3421 | . . 3 | |
22 | 20, 21 | syl 14 | . 2 |
23 | 3, 17, 22 | mpjaod 670 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wo 661 wceq 1284 wcel 1433 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: opth 3992 dmsnopg 4812 funcnvsn 4965 oprabid 5557 |
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