Theorem zfcndreg 10572

Description: Axiom of Regularity ax-reg 9537, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2402. (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 2183	. 2 ⊢ Ⅎ𝑦∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))
2		axregnd 10559	. 2 ⊢ (𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
3	1, 2	exlimi 2251	1 ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1557  ∃wex 1798   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402  ax-ext 2733  ax-sep 5245  ax-reg 9537

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-cleq 2753  df-clel 2836  df-nfc 2910

This theorem is referenced by: (None)