Theorem trsucss 2309

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 2050	. . 3 |- (Tr A -> (B e. A -> B (_ A))
2		eqimss 1548	. . . 4 |- (B = A -> B (_ A)
3	2	a1i 7	. . 3 |- (Tr A -> (B = A -> B (_ A))
4	1, 3	jaod 329	. 2 |- (Tr A -> ((B e. A \/ B = A) -> B (_ A))
5		elsuci 2289	. 2 |- (B e. suc A -> (B e. A \/ B = A))
6	4, 5	syl5 22	1 |- (Tr A -> (B e. suc A -> B (_ A))

Syntax hints:   -> wi 2   \/ wo 195   = wceq 1091   e. wcel 1092   (_ wss 1487  Tr wtr 2041  suc csuc 2201

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-uni 1920  df-tr 2042  df-suc 2205

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