Theorem zfcndreg 10033

Description: Axiom of Regularity ax-reg 9050, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfcndreg

Step	Hyp	Ref	Expression
1		nfe1 2150	. 2 ⊢ Ⅎ𝑦∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))
2		axregnd 10020	. 2 ⊢ (𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
3	1, 2	exlimi 2213	1 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1531  ∃wex 1776   ∈ wcel 2110

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-reg 9050

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4561  df-pr 4563

This theorem is referenced by: (None)