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Metamath Proof Explorer

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 ) ) )