Theorem sucssel 3070

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 2063	. 2 |- (suc A (_ B -> (A e. suc A -> A e. B))
2		sucidg 3052	. 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 958   (_ wss 2047  suc csuc 2950

This theorem is referenced by:  ordelsuc 3071  ordsucelsuc 3073  suc11 3093  oaordi 4180  unbnn2 4545  r1ord 4655  rankelun 4707  cflim 4909  indpi 5034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-in 2051  df-ss 2053  df-sn 2412  df-pr 2413  df-suc 2954

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