Theorem trsuc 2308

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 2048	. . . . . 6 ⊢ (Tr A → ((B ∈ suc B ∧ suc B ∈ A) → B ∈ A))
2	1	exp3a 292	. . . . 5 ⊢ (Tr A → (B ∈ suc B → (suc B ∈ A → B ∈ A)))
3		sucidg 2305	. . . . 5 ⊢ (B ∈ V → B ∈ suc B)
4	2, 3	syl5 22	. . . 4 ⊢ (Tr A → (B ∈ V → (suc B ∈ A → B ∈ A)))
5	4	com12 13	. . 3 ⊢ (B ∈ V → (Tr A → (suc B ∈ A → B ∈ A)))
6		sucprc 2297	. . . . . 6 ⊢ (¬ B ∈ V → suc B = B)
7	6	eleq1d 1155	. . . . 5 ⊢ (¬ B ∈ V → (suc B ∈ A ↔ B ∈ A))
8	7	biimpd 135	. . . 4 ⊢ (¬ B ∈ V → (suc B ∈ A → B ∈ A))
9	8	a1d 14	. . 3 ⊢ (¬ B ∈ V → (Tr A → (suc B ∈ A → B ∈ A)))
10	5, 9	pm2.61i 110	. 2 ⊢ (Tr A → (suc B ∈ A → B ∈ A))
11	10	imp 277	1 ⊢ ((Tr A ∧ suc B ∈ A) → B ∈ A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wcel 1092  Vcvv 1348  Tr wtr 2041  suc csuc 2201

This theorem is referenced by:  onuninsuc 2356  limsuc 2361

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-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-uni 1920  df-tr 2042  df-suc 2205

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