MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axreg2 Structured version   Visualization version   GIF version

Theorem axreg2 9049
Description: Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
axreg2 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axreg2
StepHypRef Expression
1 ax-reg 9048 . 2 (∃𝑥 𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
2119.23bi 2183 1 (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1529  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-12 2170  ax-reg 9048
This theorem depends on definitions:  df-bi 209  df-ex 1775
This theorem is referenced by:  zfregcl  9050  axregndlem2  10017
  Copyright terms: Public domain W3C validator