Theorem sucel 5798

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 3062	. 2 ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = suc 𝐴)
2		dfcleq 2616	. . . 4 ⊢ (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴))
3		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	elsuc 5794	. . . . . 6 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
5	4	bibi2i 327	. . . . 5 ⊢ ((𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴) ↔ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
6	5	albii 1747	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
7	2, 6	bitri 264	. . 3 ⊢ (𝑥 = suc 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
8	7	rexbii 3041	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = suc 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
9	1, 8	bitri 264	1 ⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))

Syntax hints:   ↔ wb 196   ∨ wo 383  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-un 3579  df-sn 4178  df-suc 5729

This theorem is referenced by:  axinf2  8537  zfinf2  8539