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Mirrors > Home > ILE Home > Th. List > preqr1g | Unicode version |
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3539. (Contributed by Jim Kingdon, 21-Sep-2018.) |
Ref | Expression |
---|---|
preqr1g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 3474 | . . . . . . 7 | |
2 | eleq2 2101 | . . . . . . 7 | |
3 | 1, 2 | syl5ibcom 144 | . . . . . 6 |
4 | elprg 3395 | . . . . . 6 | |
5 | 3, 4 | sylibd 138 | . . . . 5 |
6 | 5 | adantr 261 | . . . 4 |
7 | 6 | imp 115 | . . 3 |
8 | prid1g 3474 | . . . . . . 7 | |
9 | eleq2 2101 | . . . . . . 7 | |
10 | 8, 9 | syl5ibrcom 146 | . . . . . 6 |
11 | elprg 3395 | . . . . . 6 | |
12 | 10, 11 | sylibd 138 | . . . . 5 |
13 | 12 | adantl 262 | . . . 4 |
14 | 13 | imp 115 | . . 3 |
15 | eqcom 2042 | . . 3 | |
16 | eqeq2 2049 | . . 3 | |
17 | 7, 14, 15, 16 | oplem1 882 | . 2 |
18 | 17 | ex 108 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 wceq 1243 wcel 1393 cvv 2557 cpr 3376 |
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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 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-sn 3381 df-pr 3382 |
This theorem is referenced by: preqr2g 3538 |
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