Theorem axreg 1083

Description: Axiom of Regularity expressed more compactly.

Assertion

Ref	Expression
axreg	⊢ (x ∈ y → ∃x(x ∈ y ∧ ∀z(z ∈ x → ¬ z ∈ y)))

Distinct variable group(s):   x,y,z

Proof of Theorem axreg

Step	Hyp	Ref	Expression
1		ax-reg 1078	. 2 ⊢ (∃x x ∈ y → ∃x(x ∈ y ∧ ∀z(z ∈ x → ¬ z ∈ y)))
2	1	19.23bi 747	1 ⊢ (x ∈ y → ∃x(x ∈ y ∧ ∀z(z ∈ x → ¬ z ∈ y)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803

This theorem is referenced by:  zfregcl 3446  axregndlem2 3749

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-reg 1078

This theorem depends on definitions:  df-bi 128  df-ex 679

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