Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3597 | . . 3 | |
2 | 1 | eqeq2i 2181 | . 2 |
3 | preqsn.1 | . . . 4 | |
4 | preqsn.2 | . . . 4 | |
5 | preqsn.3 | . . . 4 | |
6 | 3, 4, 5, 5 | preq12b 3757 | . . 3 |
7 | oridm 752 | . . . 4 | |
8 | eqtr3 2190 | . . . . . 6 | |
9 | simpr 109 | . . . . . 6 | |
10 | 8, 9 | jca 304 | . . . . 5 |
11 | eqtr 2188 | . . . . . 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 703 wceq 1348 wcel 2141 cvv 2730 csn 3583 cpr 3584 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 |
This theorem is referenced by: opeqsn 4237 relop 4761 |
Copyright terms: Public domain | W3C validator |