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Mirrors > Home > ILE Home > Th. List > preq12b | Unicode version |
Description: Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.) |
Ref | Expression |
---|---|
preq12b.1 | |
preq12b.2 | |
preq12b.3 | |
preq12b.4 |
Ref | Expression |
---|---|
preq12b |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12b.1 | . . . . . 6 | |
2 | 1 | prid1 3629 | . . . . 5 |
3 | eleq2 2203 | . . . . 5 | |
4 | 2, 3 | mpbii 147 | . . . 4 |
5 | 1 | elpr 3548 | . . . 4 |
6 | 4, 5 | sylib 121 | . . 3 |
7 | preq1 3600 | . . . . . . . 8 | |
8 | 7 | eqeq1d 2148 | . . . . . . 7 |
9 | preq12b.2 | . . . . . . . 8 | |
10 | preq12b.4 | . . . . . . . 8 | |
11 | 9, 10 | preqr2 3696 | . . . . . . 7 |
12 | 8, 11 | syl6bi 162 | . . . . . 6 |
13 | 12 | com12 30 | . . . . 5 |
14 | 13 | ancld 323 | . . . 4 |
15 | prcom 3599 | . . . . . . 7 | |
16 | 15 | eqeq2i 2150 | . . . . . 6 |
17 | preq1 3600 | . . . . . . . . 9 | |
18 | 17 | eqeq1d 2148 | . . . . . . . 8 |
19 | preq12b.3 | . . . . . . . . 9 | |
20 | 9, 19 | preqr2 3696 | . . . . . . . 8 |
21 | 18, 20 | syl6bi 162 | . . . . . . 7 |
22 | 21 | com12 30 | . . . . . 6 |
23 | 16, 22 | sylbi 120 | . . . . 5 |
24 | 23 | ancld 323 | . . . 4 |
25 | 14, 24 | orim12d 775 | . . 3 |
26 | 6, 25 | mpd 13 | . 2 |
27 | preq12 3602 | . . 3 | |
28 | prcom 3599 | . . . . 5 | |
29 | 17, 28 | syl6eq 2188 | . . . 4 |
30 | preq1 3600 | . . . 4 | |
31 | 29, 30 | sylan9eq 2192 | . . 3 |
32 | 27, 31 | jaoi 705 | . 2 |
33 | 26, 32 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 cvv 2686 cpr 3528 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 |
This theorem is referenced by: prel12 3698 opthpr 3699 preq12bg 3700 preqsn 3702 opeqpr 4175 preleq 4470 |
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