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Mirrors > Home > ILE Home > Th. List > preleq | Unicode version |
Description: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Ref | Expression |
---|---|
preleq.1 | |
preleq.2 | |
preleq.3 | |
preleq.4 |
Ref | Expression |
---|---|
preleq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2lp 4538 | . . . . 5 | |
2 | eleq12 2235 | . . . . . 6 | |
3 | 2 | anbi1d 462 | . . . . 5 |
4 | 1, 3 | mtbiri 670 | . . . 4 |
5 | 4 | con2i 622 | . . 3 |
6 | 5 | adantr 274 | . 2 |
7 | preleq.1 | . . . . 5 | |
8 | preleq.2 | . . . . 5 | |
9 | preleq.3 | . . . . 5 | |
10 | preleq.4 | . . . . 5 | |
11 | 7, 8, 9, 10 | preq12b 3757 | . . . 4 |
12 | 11 | biimpi 119 | . . 3 |
13 | 12 | adantl 275 | . 2 |
14 | 6, 13 | ecased 1344 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 wceq 1348 wcel 2141 cvv 2730 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-in1 609 ax-in2 610 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 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-dif 3123 df-un 3125 df-sn 3589 df-pr 3590 |
This theorem is referenced by: opthreg 4540 |
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