Theorem sucel 6439

Description: Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem sucel

Step	Hyp	Ref	Expression
1		risset 3237	. 2 ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = suc 𝐴)
2		dfcleq 2753	. . . 4 ⊢ (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴))
3		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	elsuc 6435	. . . . . 6 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
5	4	bibi2i 340	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
6	5	albii 1851	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
7	2, 6	bitri 278	. . 3 ⊢ (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
8	7	rexbii 3109	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
9	1, 8	bitri 278	1 ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-v 3452  df-un 4151  df-sn 4585  df-suc 6368

This theorem is used by:  axinf2  9644  zfinf2  9646