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Mirrors > Home > ILE Home > Th. List > opeluu | Unicode version |
Description: Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
opeluu.1 | |
opeluu.2 |
Ref | Expression |
---|---|
opeluu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeluu.1 | . . . 4 | |
2 | 1 | prid1 3599 | . . 3 |
3 | opeluu.2 | . . . . 5 | |
4 | 1, 3 | opi2 4125 | . . . 4 |
5 | elunii 3711 | . . . 4 | |
6 | 4, 5 | mpan 420 | . . 3 |
7 | elunii 3711 | . . 3 | |
8 | 2, 6, 7 | sylancr 410 | . 2 |
9 | 3 | prid2 3600 | . . 3 |
10 | elunii 3711 | . . 3 | |
11 | 9, 6, 10 | sylancr 410 | . 2 |
12 | 8, 11 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 cvv 2660 cpr 3498 cop 3500 cuni 3706 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 |
This theorem is referenced by: asymref 4894 |
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