Theorem trsuc 3055

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

Assertion

Ref	Expression
trsuc	⊢ ((Tr A ⋀ suc B ∈ A) → B ∈ A)

Proof of Theorem trsuc

Step	Hyp	Ref	Expression
1		trel 2683	. . . . 5 ⊢ (Tr A → ((B ∈ suc B ⋀ suc B ∈ A) → B ∈ A))
2	1	exp3a 375	. . . 4 ⊢ (Tr A → (B ∈ suc B → (suc B ∈ A → B ∈ A)))
3		sucidg 3052	. . . 4 ⊢ (B ∈ V → B ∈ suc B)
4	2, 3	syl5com 52	. . 3 ⊢ (B ∈ V → (Tr A → (suc B ∈ A → B ∈ A)))
5		sucprc 3044	. . . . . 6 ⊢ (¬ B ∈ V → suc B = B)
6	5	eleq1d 1537	. . . . 5 ⊢ (¬ B ∈ V → (suc B ∈ A ↔ B ∈ A))
7	6	biimpd 153	. . . 4 ⊢ (¬ B ∈ V → (suc B ∈ A → B ∈ A))
8	7	a1d 12	. . 3 ⊢ (¬ B ∈ V → (Tr A → (suc B ∈ A → B ∈ A)))
9	4, 8	pm2.61i 126	. 2 ⊢ (Tr A → (suc B ∈ A → B ∈ A))
10	9	imp 350	1 ⊢ ((Tr A ⋀ suc B ∈ A) → B ∈ A)

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   ∈ wcel 956  Vcvv 1807  Tr wtr 2676  suc csuc 2949

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-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409  df-uni 2500  df-tr 2677  df-suc 2953

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