Theorem n0el 4311

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 3071	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
2		df-ex 1794	. . 3 ⊢ (∃𝑢 𝑢 ∈ 𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)
3	2	ralbii 3102	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)
4		alnex 1795	. . 3 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
5		imnang 1856	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
6		0el 4310	. . . . 5 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥)
7		df-rex 3081	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
8	6, 7	bitri 277	. . . 4 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
9	8	notbii 322	. . 3 ⊢ (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
10	4, 5, 9	3bitr4ri 306	. 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
11	1, 3, 10	3bitr4ri 306	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1552  ∃wex 1793   ∈ wcel 2136  ∀wral 3070  ∃wrex 3080  ∅c0 4280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-dif 3902  df-nul 4281

This theorem is referenced by:  n0el2  38782  prter2  39453