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Mirrors > Home > ILE Home > Th. List > abnex | Unicode version |
Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 4420 and pwnex 4421. See the comment of abnexg 4418. (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
abnex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 4108 | . 2 | |
2 | alral 2509 | . . 3 | |
3 | rexv 2739 | . . . . . . 7 | |
4 | 3 | bicomi 131 | . . . . . 6 |
5 | 4 | abbii 2280 | . . . . 5 |
6 | 5 | eleq1i 2230 | . . . 4 |
7 | 6 | biimpi 119 | . . 3 |
8 | abnexg 4418 | . . 3 | |
9 | 2, 7, 8 | syl2im 38 | . 2 |
10 | 1, 9 | mtoi 654 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wal 1340 wceq 1342 wex 1479 wcel 2135 cab 2150 wral 2442 wrex 2443 cvv 2721 |
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 604 ax-in2 605 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-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-in 3117 df-ss 3124 df-sn 3576 df-uni 3784 df-iun 3862 |
This theorem is referenced by: pwnex 4421 |
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