Theorem trsucss 3062

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 2694	. . 3 |- (Tr A -> (B e. A -> B (_ A))
2		eqimss 2112	. . . 4 |- (B = A -> B (_ A)
3	2	a1i 8	. . 3 |- (Tr A -> (B = A -> B (_ A))
4	1, 3	jaod 426	. 2 |- (Tr A -> ((B e. A \/ B = A) -> B (_ A))
5		elsuci 3041	. 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 958   e. wcel 960   (_ wss 2050  Tr wtr 2685  suc csuc 2956

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-sn 2416  df-pr 2417  df-uni 2508  df-tr 2686  df-suc 2960

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