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Mirrors > Home > ILE Home > Th. List > opeqsn | Unicode version |
Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
Ref | Expression |
---|---|
opeqsn.1 | |
opeqsn.2 | |
opeqsn.3 |
Ref | Expression |
---|---|
opeqsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeqsn.1 | . . . 4 | |
2 | opeqsn.2 | . . . 4 | |
3 | 1, 2 | dfop 3699 | . . 3 |
4 | 3 | eqeq1i 2145 | . 2 |
5 | 1 | snex 4104 | . . 3 |
6 | prexg 4128 | . . . 4 | |
7 | 1, 2, 6 | mp2an 422 | . . 3 |
8 | opeqsn.3 | . . 3 | |
9 | 5, 7, 8 | preqsn 3697 | . 2 |
10 | eqcom 2139 | . . . . 5 | |
11 | 1, 2, 1 | preqsn 3697 | . . . . 5 |
12 | eqcom 2139 | . . . . . . 7 | |
13 | 12 | anbi2i 452 | . . . . . 6 |
14 | anidm 393 | . . . . . 6 | |
15 | 13, 14 | bitri 183 | . . . . 5 |
16 | 10, 11, 15 | 3bitri 205 | . . . 4 |
17 | 16 | anbi1i 453 | . . 3 |
18 | dfsn2 3536 | . . . . . . 7 | |
19 | preq2 3596 | . . . . . . 7 | |
20 | 18, 19 | syl5req 2183 | . . . . . 6 |
21 | 20 | eqeq1d 2146 | . . . . 5 |
22 | eqcom 2139 | . . . . 5 | |
23 | 21, 22 | syl6bb 195 | . . . 4 |
24 | 23 | pm5.32i 449 | . . 3 |
25 | 17, 24 | bitri 183 | . 2 |
26 | 4, 9, 25 | 3bitri 205 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wcel 1480 cvv 2681 csn 3522 cpr 3523 cop 3525 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 |
This theorem is referenced by: relop 4684 |
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