Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > preqr1 | Unicode version |
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.) |
Ref | Expression |
---|---|
preqr1.1 | |
preqr1.2 |
Ref | Expression |
---|---|
preqr1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preqr1.1 | . . . . 5 | |
2 | 1 | prid1 3476 | . . . 4 |
3 | eleq2 2101 | . . . 4 | |
4 | 2, 3 | mpbii 136 | . . 3 |
5 | 1 | elpr 3396 | . . 3 |
6 | 4, 5 | sylib 127 | . 2 |
7 | preqr1.2 | . . . . 5 | |
8 | 7 | prid1 3476 | . . . 4 |
9 | eleq2 2101 | . . . 4 | |
10 | 8, 9 | mpbiri 157 | . . 3 |
11 | 7 | elpr 3396 | . . 3 |
12 | 10, 11 | sylib 127 | . 2 |
13 | eqcom 2042 | . 2 | |
14 | eqeq2 2049 | . 2 | |
15 | 6, 12, 13, 14 | oplem1 882 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 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: preqr2 3540 |
Copyright terms: Public domain | W3C validator |