Theorem trsuc 3055

Description: A set whose successor belongs to a transitive class also belongs.

Assertion

Ref	Expression
trsuc	|- ((Tr A /\ suc B e. A) -> B e. A)

Proof of Theorem trsuc

Step	Hyp	Ref	Expression
1		trel 2687	. . . . 5 |- (Tr A -> ((B e. suc B /\ suc B e. A) -> B e. A))
2	1	exp3a 375	. . . 4 |- (Tr A -> (B e. suc B -> (suc B e. A -> B e. A)))
3		sucidg 3052	. . . 4 |- (B e. V -> B e. suc B)
4	2, 3	syl5com 52	. . 3 |- (B e. V -> (Tr A -> (suc B e. A -> B e. A)))
5		sucprc 3044	. . . . . 6 |- (-. B e. V -> suc B = B)
6	5	eleq1d 1540	. . . . 5 |- (-. B e. V -> (suc B e. A <-> B e. A))
7	6	biimpd 153	. . . 4 |- (-. B e. V -> (suc B e. A -> B e. A))
8	7	a1d 12	. . 3 |- (-. B e. V -> (Tr A -> (suc B e. A -> B e. A)))
9	4, 8	pm2.61i 126	. 2 |- (Tr A -> (suc B e. A -> B e. A))
10	9	imp 350	1 |- ((Tr A /\ suc B e. A) -> B e. A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 958  Vcvv 1811  Tr wtr 2680  suc csuc 2950

This theorem is referenced by:  onuninsuc 3108  limsuc 3120

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-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-uni 2504  df-tr 2681  df-suc 2954

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