Theorem trsuc 3018

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 2655	. . . . 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 3015	. . . 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 3007	. . . . . 6 |- (-. B e. V -> suc B = B)
6	5	eleq1d 1516	. . . . 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 1105  Vcvv 1786  Tr wtr 2648  suc csuc 2913

This theorem is referenced by:  onuninsuc 3071  limsuc 3083

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-sn 2383  df-pr 2384  df-uni 2472  df-tr 2649  df-suc 2917

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