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Theorem zfcndreg 10091
 Description: Axiom of Regularity ax-reg 9103, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2380. (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 2152 . 2 𝑦𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥))
2 axregnd 10078 . 2 (𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
31, 2exlimi 2216 1 (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1537  ∃wex 1782   ∈ wcel 2112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-reg 9103 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3864  df-un 3866  df-nul 4229  df-sn 4527  df-pr 4529 This theorem is referenced by: (None)
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