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Theorem zfcndreg 10660
Description: Axiom of Regularity ax-reg 9635, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2366. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
zfcndreg (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfcndreg
StepHypRef Expression
1 nfe1 2140 . 2 𝑦𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥))
2 axregnd 10647 . 2 (𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
31, 2exlimi 2206 1 (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wal 1532  wex 1774  wcel 2099
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2366  ax-ext 2697  ax-sep 5304  ax-pr 5433  ax-reg 9635
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-v 3464  df-un 3952  df-sn 4634  df-pr 4636
This theorem is referenced by: (None)
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