Theorem axreg2 9640

Description: Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)

Assertion

Ref	Expression
axreg2	⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axreg2

Step	Hyp	Ref	Expression
1		ax-reg 9639	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
2	1	19.23bi 2191	1 ⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177  ax-reg 9639

This theorem depends on definitions:  df-bi 207  df-ex 1779

This theorem is referenced by:  zfregcl  9641  axregndlem2  10650