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Mirrors > Home > ILE Home > Th. List > ssoprab2 | Unicode version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4030. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
ssoprab2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . . . . . . 10 | |
2 | 1 | anim2d 330 | . . . . . . . . 9 |
3 | 2 | alimi 1384 | . . . . . . . 8 |
4 | exim 1530 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 5 | alimi 1384 | . . . . . 6 |
7 | exim 1530 | . . . . . 6 | |
8 | 6, 7 | syl 14 | . . . . 5 |
9 | 8 | alimi 1384 | . . . 4 |
10 | exim 1530 | . . . 4 | |
11 | 9, 10 | syl 14 | . . 3 |
12 | 11 | ss2abdv 3067 | . 2 |
13 | df-oprab 5536 | . 2 | |
14 | df-oprab 5536 | . 2 | |
15 | 12, 13, 14 | 3sstr4g 3040 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wal 1282 wceq 1284 wex 1421 cab 2067 wss 2973 cop 3401 coprab 5533 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-in 2979 df-ss 2986 df-oprab 5536 |
This theorem is referenced by: ssoprab2b 5582 |
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