Theorem sucidg 3047

Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized).

Assertion

Ref	Expression
sucidg	|- (A e. B -> A e. suc A)

Proof of Theorem sucidg

Step	Hyp	Ref	Expression
1		id 59	. . 3 |- (x = A -> x = A)
2		suceq 3029	. . 3 |- (x = A -> suc x = suc A)
3	1, 2	eleq12d 1539	. 2 |- (x = A -> (x e. suc x <-> A e. suc A))
4		visset 1809	. . 3 |- x e. V
5	4	sucid 3046	. 2 |- x e. suc x
6	3, 5	vtoclg 1843	1 |- (A e. B -> A e. suc A)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  suc csuc 2945

This theorem is referenced by:  nsuceq0 3048  trsuc 3050  ordsuc 3060  sucssel 3065  onsucuni2 3086  nlimsucg 3107  tfrlem13 3914  oarec 4186  oeordi 4204  php4 4502  suc11reg 4585

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-sn 2408  df-pr 2409  df-suc 2949

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