Theorem n0el 3973

Description: Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Assertion

Ref	Expression
n0el	⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑢

Allowed substitution hint:   𝐴(𝑢)

Proof of Theorem n0el

Step	Hyp	Ref	Expression
1		df-ral 2946	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
2		df-ex 1745	. . 3 ⊢ (∃𝑢 𝑢 ∈ 𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)
3	2	ralbii 3009	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)
4		alnex 1746	. . 3 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
5		imnang 1809	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
6		0el 3972	. . . . 5 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥)
7		df-rex 2947	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
8	6, 7	bitri 264	. . . 4 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
9	8	notbii 309	. . 3 ⊢ (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
10	4, 5, 9	3bitr4ri 293	. 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
11	1, 3, 10	3bitr4ri 293	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521  ∃wex 1744   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  ∅c0 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-nul 3949

This theorem is referenced by:  n0el2  34244  prter2  34485