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Theorem sucprcreg 8046
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.)
Assertion
Ref Expression
sucprcreg

Proof of Theorem sucprcreg
StepHypRef Expression
1 sucprc 4958 . 2
2 elirr 8045 . . . 4
3 df-suc 4889 . . . . . . . 8
43eqeq1i 2464 . . . . . . 7
5 ssequn2 3676 . . . . . . 7
64, 5bitr4i 252 . . . . . 6
76biimpi 194 . . . . 5
8 snidg 4055 . . . . 5
9 ssel2 3498 . . . . 5
107, 8, 9syl2an 477 . . . 4
112, 10mto 176 . . 3
1211imnani 423 . 2
131, 12impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  u.cun 3473  C_wss 3475  {csn 4029  succsuc 4885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-reg 8039
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-pr 4032  df-suc 4889
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