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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 3531 | . . 3 |
3 | 2 | a1i 9 | . 2 |
4 | opth1.2 | . . . . . . . . 9 | |
5 | 1, 4 | opi1 3969 | . . . . . . . 8 |
6 | id 19 | . . . . . . . 8 | |
7 | 5, 6 | syl5eleq 2126 | . . . . . . 7 |
8 | oprcl 3573 | . . . . . . 7 | |
9 | 7, 8 | syl 14 | . . . . . 6 |
10 | 9 | simpld 105 | . . . . 5 |
11 | prid1g 3474 | . . . . 5 | |
12 | 10, 11 | syl 14 | . . . 4 |
13 | eleq2 2101 | . . . 4 | |
14 | 12, 13 | syl5ibrcom 146 | . . 3 |
15 | elsni 3393 | . . . 4 | |
16 | 15 | eqcomd 2045 | . . 3 |
17 | 14, 16 | syl6 29 | . 2 |
18 | dfopg 3547 | . . . . 5 | |
19 | 7, 8, 18 | 3syl 17 | . . . 4 |
20 | 7, 19 | eleqtrd 2116 | . . 3 |
21 | elpri 3398 | . . 3 | |
22 | 20, 21 | syl 14 | . 2 |
23 | 3, 17, 22 | mpjaod 638 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 wceq 1243 wcel 1393 cvv 2557 csn 3375 cpr 3376 cop 3378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 |
This theorem is referenced by: opth 3974 dmsnopg 4792 funcnvsn 4945 oprabid 5537 |
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