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Mirrors > Home > ILE Home > Th. List > abnexg | Unicode version |
Description: Sufficient condition for a class abstraction to be a proper class. The class can be thought of as an expression in and the abstraction appearing in the statement as the class of values as varies through . Assuming the antecedents, if that class is a set, then so is the "domain" . The converse holds without antecedent, see abrexexg 6097. Note that the second antecedent cannot be translated to since may depend on . In applications, one may take or (see snnex 4433 and pwnex 4434 respectively, proved from abnex 4432, which is a consequence of abnexg 4431 with ). (Contributed by BJ, 2-Dec-2021.) |
Ref | Expression |
---|---|
abnexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 4424 | . 2 | |
2 | simpl 108 | . . . . 5 | |
3 | 2 | ralimi 2533 | . . . 4 |
4 | dfiun2g 3905 | . . . . . 6 | |
5 | 4 | eleq1d 2239 | . . . . 5 |
6 | 5 | biimprd 157 | . . . 4 |
7 | 3, 6 | syl 14 | . . 3 |
8 | simpr 109 | . . . . 5 | |
9 | 8 | ralimi 2533 | . . . 4 |
10 | iunid 3928 | . . . . 5 | |
11 | snssi 3724 | . . . . . . 7 | |
12 | 11 | ralimi 2533 | . . . . . 6 |
13 | ss2iun 3888 | . . . . . 6 | |
14 | 12, 13 | syl 14 | . . . . 5 |
15 | 10, 14 | eqsstrrid 3194 | . . . 4 |
16 | ssexg 4128 | . . . . 5 | |
17 | 16 | ex 114 | . . . 4 |
18 | 9, 15, 17 | 3syl 17 | . . 3 |
19 | 7, 18 | syld 45 | . 2 |
20 | 1, 19 | syl5 32 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cab 2156 wral 2448 wrex 2449 cvv 2730 wss 3121 csn 3583 cuni 3796 ciun 3873 |
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 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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-un 4418 |
This theorem depends on definitions: df-bi 116 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-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-sn 3589 df-uni 3797 df-iun 3875 |
This theorem is referenced by: abnex 4432 |
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