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Mirrors > Home > MPE Home > Th. List > abnexg | Structured version Visualization version 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 7140. Note that the second antecedent cannot be translated to since may depend on . In applications, one may take or (see snnex 6966 and pwnex 6968 respectively, proved from abnex 6965, which is a consequence of abnexg 6964 with ). (Contributed by BJ, 2-Dec-2021.) |
Ref | Expression |
---|---|
abnexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 6955 | . 2 | |
2 | simpl 473 | . . . . 5 | |
3 | 2 | ralimi 2952 | . . . 4 |
4 | dfiun2g 4552 | . . . . . 6 | |
5 | 4 | eleq1d 2686 | . . . . 5 |
6 | 5 | biimprd 238 | . . . 4 |
7 | 3, 6 | syl 17 | . . 3 |
8 | simpr 477 | . . . . 5 | |
9 | 8 | ralimi 2952 | . . . 4 |
10 | iunid 4575 | . . . . 5 | |
11 | snssi 4339 | . . . . . . 7 | |
12 | 11 | ralimi 2952 | . . . . . 6 |
13 | ss2iun 4536 | . . . . . 6 | |
14 | 12, 13 | syl 17 | . . . . 5 |
15 | 10, 14 | syl5eqssr 3650 | . . . 4 |
16 | ssexg 4804 | . . . . 5 | |
17 | 16 | ex 450 | . . . 4 |
18 | 9, 15, 17 | 3syl 18 | . . 3 |
19 | 7, 18 | syld 47 | . 2 |
20 | 1, 19 | syl5 34 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cab 2608 wral 2912 wrex 2913 cvv 3200 wss 3574 csn 4177 cuni 4436 ciun 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-in 3581 df-ss 3588 df-sn 4178 df-uni 4437 df-iun 4522 |
This theorem is referenced by: abnex 6965 |
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