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Mirrors > Home > ILE Home > Th. List > preqsn | Unicode version |
Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
Ref | Expression |
---|---|
preqsn.1 | |
preqsn.2 | |
preqsn.3 |
Ref | Expression |
---|---|
preqsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3574 | . . 3 | |
2 | 1 | eqeq2i 2168 | . 2 |
3 | preqsn.1 | . . . 4 | |
4 | preqsn.2 | . . . 4 | |
5 | preqsn.3 | . . . 4 | |
6 | 3, 4, 5, 5 | preq12b 3733 | . . 3 |
7 | oridm 747 | . . . 4 | |
8 | eqtr3 2177 | . . . . . 6 | |
9 | simpr 109 | . . . . . 6 | |
10 | 8, 9 | jca 304 | . . . . 5 |
11 | eqtr 2175 | . . . . . 6 | |
12 | simpr 109 | . . . . . 6 | |
13 | 11, 12 | jca 304 | . . . . 5 |
14 | 10, 13 | impbii 125 | . . . 4 |
15 | 7, 14 | bitri 183 | . . 3 |
16 | 6, 15 | bitri 183 | . 2 |
17 | 2, 16 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wo 698 wceq 1335 wcel 2128 cvv 2712 csn 3560 cpr 3561 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-sn 3566 df-pr 3567 |
This theorem is referenced by: opeqsn 4212 relop 4736 |
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