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Theorem zfcndreg 10655
Description: Axiom of Regularity ax-reg 9630, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2375. (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 2148 . 2 𝑦𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥))
2 axregnd 10642 . 2 (𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
31, 2exlimi 2215 1 (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  wex 1776  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by: (None)
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