Theorem n0el 4354

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 3054	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
2		df-ex 1774	. . 3 ⊢ (∃𝑢 𝑢 ∈ 𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)
3	2	ralbii 3085	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)
4		alnex 1775	. . 3 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
5		imnang 1836	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
6		0el 4353	. . . . 5 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥)
7		df-rex 3063	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
8	6, 7	bitri 275	. . . 4 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
9	8	notbii 320	. . 3 ⊢ (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
10	4, 5, 9	3bitr4ri 304	. 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
11	1, 3, 10	3bitr4ri 304	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531  ∃wex 1773   ∈ wcel 2098  ∀wral 3053  ∃wrex 3062  ∅c0 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-dif 3944  df-nul 4316

This theorem is referenced by:  n0el2  37696  prter2  38245