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Theorem sucprcreg 8114
 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 5498 . 2
2 elirr 8113 . . . 4
3 df-suc 5429 . . . . . . . 8
43eqeq1i 2456 . . . . . . 7
5 ssequn2 3607 . . . . . . 7
64, 5bitr4i 256 . . . . . 6
76biimpi 198 . . . . 5
8 snidg 3994 . . . . 5
9 ssel2 3427 . . . . 5
107, 8, 9syl2an 480 . . . 4
112, 10mto 180 . . 3
1211imnani 425 . 2
131, 12impbii 191 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 188   wa 371   wceq 1444   wcel 1887  cvv 3045   cun 3402   wss 3404  csn 3968   csuc 5425 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-reg 8107 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969  df-pr 3971  df-suc 5429 This theorem is referenced by: (None)
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