Theorem axpowndlem1

Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002)

	Ref	Expression
Assertion	axpowndlem1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑦 → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.24	⊢ ( 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑦 → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) )
2	1	sps	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑦 → ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( ∃ 𝑧 𝑥 ∈ 𝑦 → ∀ 𝑦 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) )