Theorem axpowndlem1 3743

Description: Lemma for the Axiom of Power Sets with no distinct variable conditions.

Assertion

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

Proof of Theorem axpowndlem1

Step	Hyp	Ref	Expression
1		pm2.24 72	. 2 ⊢ (x = y → (¬ x = y → ∃x∀y(∀x(∃z x ∈ y → ∀y x ∈ z) → y ∈ x)))
2	1	a4s 682	1 ⊢ (∀x x = y → (¬ x = y → ∃x∀y(∀x(∃z x ∈ y → ∀y x ∈ z) → y ∈ x)))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803

This theorem is referenced by:  axpowndlem3 3745  axpownd 3747

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673

metamath.org