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Mirrors > Home > ILE Home > Th. List > ssoprab2 | Unicode version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4248. (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 335 | . . . . . . . . 9 |
3 | 2 | alimi 1442 | . . . . . . . 8 |
4 | exim 1586 | . . . . . . . 8 | |
5 | 3, 4 | syl 14 | . . . . . . 7 |
6 | 5 | alimi 1442 | . . . . . 6 |
7 | exim 1586 | . . . . . 6 | |
8 | 6, 7 | syl 14 | . . . . 5 |
9 | 8 | alimi 1442 | . . . 4 |
10 | exim 1586 | . . . 4 | |
11 | 9, 10 | syl 14 | . . 3 |
12 | 11 | ss2abdv 3211 | . 2 |
13 | df-oprab 5841 | . 2 | |
14 | df-oprab 5841 | . 2 | |
15 | 12, 13, 14 | 3sstr4g 3181 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1340 wceq 1342 wex 1479 cab 2150 wss 3112 cop 3574 coprab 5838 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-in 3118 df-ss 3125 df-oprab 5841 |
This theorem is referenced by: ssoprab2b 5891 |
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