Theorem n0el 4275

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 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
2		df-ex 1782	. . 3 ⊢ (∃𝑢 𝑢 ∈ 𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)
3	2	ralbii 3133	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)
4		alnex 1783	. . 3 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
5		imnang 1843	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
6		0el 4274	. . . . 5 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥)
7		df-rex 3112	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
8	6, 7	bitri 278	. . . 4 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
9	8	notbii 323	. . 3 ⊢ (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
10	4, 5, 9	3bitr4ri 307	. 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥))
11	1, 3, 10	3bitr4ri 307	1 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-dif 3884  df-nul 4244

This theorem is referenced by:  n0el2  35750  prter2  36177