Theorem sucel 2296

Description: Membership of a successor in another class.

Assertion

Ref	Expression
sucel	⊢ (suc A ∈ B ↔ ∃x ∈ B ∀y(y ∈ x ↔ (y ∈ A ∨ y = A)))

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem sucel

Step	Hyp	Ref	Expression
1		risset 1235	. 2 ⊢ (suc A ∈ B ↔ ∃x ∈ B x = suc A)
2		dfcleq 1098	. . . 4 ⊢ (x = suc A ↔ ∀y(y ∈ x ↔ y ∈ suc A))
3		visset 1350	. . . . . . 7 ⊢ y ∈ V
4	3	elsuc 2292	. . . . . 6 ⊢ (y ∈ suc A ↔ (y ∈ A ∨ y = A))
5	4	bibi2i 460	. . . . 5 ⊢ ((y ∈ x ↔ y ∈ suc A) ↔ (y ∈ x ↔ (y ∈ A ∨ y = A)))
6	5	bial 695	. . . 4 ⊢ (∀y(y ∈ x ↔ y ∈ suc A) ↔ ∀y(y ∈ x ↔ (y ∈ A ∨ y = A)))
7	2, 6	bitr 151	. . 3 ⊢ (x = suc A ↔ ∀y(y ∈ x ↔ (y ∈ A ∨ y = A)))
8	7	birex 1224	. 2 ⊢ (∃x ∈ B x = suc A ↔ ∃x ∈ B ∀y(y ∈ x ↔ (y ∈ A ∨ y = A)))
9	1, 8	bitr 151	1 ⊢ (suc A ∈ B ↔ ∃x ∈ B ∀y(y ∈ x ↔ (y ∈ A ∨ y = A)))

Syntax hints:   ↔ wb 127   ∨ wo 195  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  suc csuc 2201

This theorem is referenced by:  inf4 3473  zfinf 3474

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-rex 1206  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-suc 2205

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