Theorem trsucss 3056

Description: A member of the successor of a transitive class is a subclass of it.

Assertion

Ref	Expression
trsucss	|- (Tr A -> (B e. suc A -> B (_ A))

Proof of Theorem trsucss

Step	Hyp	Ref	Expression
1		trss 2689	. . 3 |- (Tr A -> (B e. A -> B (_ A))
2		eqimss 2109	. . . 4 |- (B = A -> B (_ A)
3	2	a1i 8	. . 3 |- (Tr A -> (B = A -> B (_ A))
4	1, 3	jaod 424	. 2 |- (Tr A -> ((B e. A \/ B = A) -> B (_ A))
5		elsuci 3035	. 2 |- (B e. suc A -> (B e. A \/ B = A))
6	4, 5	syl5 21	1 |- (Tr A -> (B e. suc A -> B (_ A))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 956   e. wcel 958   (_ wss 2047  Tr wtr 2680  suc csuc 2950

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-ral 1649  df-v 1812  df-un 2050  df-in 2051  df-ss 2053  df-sn 2412  df-pr 2413  df-uni 2504  df-tr 2681  df-suc 2954

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