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Theorem abnexg 4367
 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 6016. Note that the second antecedent cannot be translated to since may depend on . In applications, one may take or (see snnex 4369 and pwnex 4370 respectively, proved from abnex 4368, which is a consequence of abnexg 4367 with ). (Contributed by BJ, 2-Dec-2021.)
Assertion
Ref Expression
abnexg
Distinct variable groups:   ,,   ,
Allowed substitution hints:   ()   (,)   (,)

Proof of Theorem abnexg
StepHypRef Expression
1 uniexg 4361 . 2
2 simpl 108 . . . . 5
32ralimi 2495 . . . 4
4 dfiun2g 3845 . . . . . 6
54eleq1d 2208 . . . . 5
65biimprd 157 . . . 4
73, 6syl 14 . . 3
8 simpr 109 . . . . 5
98ralimi 2495 . . . 4
10 iunid 3868 . . . . 5
11 snssi 3664 . . . . . . 7
1211ralimi 2495 . . . . . 6
13 ss2iun 3828 . . . . . 6
1412, 13syl 14 . . . . 5
1510, 14eqsstrrid 3144 . . . 4
16 ssexg 4067 . . . . 5
1716ex 114 . . . 4
189, 15, 173syl 17 . . 3
197, 18syld 45 . 2
201, 19syl5 32 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1331   wcel 1480  cab 2125  wral 2416  wrex 2417  cvv 2686   wss 3071  csn 3527  cuni 3736  ciun 3813 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533  df-uni 3737  df-iun 3815 This theorem is referenced by:  abnex  4368
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