Theorem sucel 6418

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 3236	. 2 ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = suc 𝐴)
2		dfcleq 2754	. . . 4 ⊢ (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴))
3		vex 3457	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	elsuc 6414	. . . . . 6 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
5	4	bibi2i 339	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
6	5	albii 1838	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
7	2, 6	bitri 277	. . 3 ⊢ (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
8	7	rexbii 3108	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
9	1, 8	bitri 277	1 ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))

Syntax hints:   ↔ wb 208   ∨ wo 858  ∀wal 1557   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  suc csuc 6344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3086  df-v 3455  df-un 3909  df-sn 4582  df-suc 6348

This theorem is referenced by:  axinf2  9592  zfinf2  9594