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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 6066. Note that the second antecedent cannot be translated to since may depend on . In applications, one may take or (see snnex 4408 and pwnex 4409 respectively, proved from abnex 4407, which is a consequence of abnexg 4406 with ). (Contributed by BJ, 2-Dec-2021.) |
Ref | Expression |
---|---|
abnexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 4399 | . 2 | |
2 | simpl 108 | . . . . 5 | |
3 | 2 | ralimi 2520 | . . . 4 |
4 | dfiun2g 3881 | . . . . . 6 | |
5 | 4 | eleq1d 2226 | . . . . 5 |
6 | 5 | biimprd 157 | . . . 4 |
7 | 3, 6 | syl 14 | . . 3 |
8 | simpr 109 | . . . . 5 | |
9 | 8 | ralimi 2520 | . . . 4 |
10 | iunid 3904 | . . . . 5 | |
11 | snssi 3700 | . . . . . . 7 | |
12 | 11 | ralimi 2520 | . . . . . 6 |
13 | ss2iun 3864 | . . . . . 6 | |
14 | 12, 13 | syl 14 | . . . . 5 |
15 | 10, 14 | eqsstrrid 3175 | . . . 4 |
16 | ssexg 4103 | . . . . 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 1335 wcel 2128 cab 2143 wral 2435 wrex 2436 cvv 2712 wss 3102 csn 3560 cuni 3772 ciun 3849 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-in 3108 df-ss 3115 df-sn 3566 df-uni 3773 df-iun 3851 |
This theorem is referenced by: abnex 4407 |
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