Theorem sucel

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

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

Proof

Step	Hyp	Ref	Expression
1		risset	⊢ ( suc 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 𝑥 = suc 𝐴 )
2		dfcleq	⊢ ( 𝑥 = suc 𝐴 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴 ) )
3		vex	⊢ 𝑦 ∈ V
4	3	elsuc	⊢ ( 𝑦 ∈ suc 𝐴 ↔ ( 𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴 ) )
5	4	bibi2i	⊢ ( ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴 ) ↔ ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴 ) ) )
6	5	albii	⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ suc 𝐴 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴 ) ) )
7	2 6	bitri	⊢ ( 𝑥 = suc 𝐴 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴 ) ) )
8	7	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐵 𝑥 = suc 𝐴 ↔ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴 ) ) )
9	1 8	bitri	⊢ ( suc 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴 ) ) )