Theorem sucssel 3065

Description: A set whose successor is a subset of another class is a member of that class.

Assertion

Ref	Expression
sucssel	|- (A e. C -> (suc A (_ B -> A e. B))

Proof of Theorem sucssel

Step	Hyp	Ref	Expression
1		ssel 2059	. 2 |- (suc A (_ B -> (A e. suc A -> A e. B))
2		sucidg 3047	. 2 |- (A e. C -> A e. suc A)
3	1, 2	syl5com 52	1 |- (A e. C -> (suc A (_ B -> A e. B))

Syntax hints:   -> wi 3   e. wcel 956   (_ wss 2043  suc csuc 2945

This theorem is referenced by:  ordelsuc 3066  ordsucelsuc 3068  suc11 3088  oaordi 4170  unbnn2 4528  r1ord 4635  rankelun 4687  cflim 4889  indpi 5014

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-in 2047  df-ss 2049  df-sn 2408  df-pr 2409  df-suc 2949

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