Theorem n0fOLD 3887

Description: Obsolete proof of n0f 3886 as of 15-Jul-2021. (Contributed by NM, 17-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
eq0f.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
n0fOLD	⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Proof of Theorem n0fOLD

Step	Hyp	Ref	Expression
1		eq0f.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
2		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑥∅
3	1, 2	cleqf 2776	. . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
4		noel 3878	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
5	4	nbn 361	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
6	5	albii 1737	. . . 4 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
7	3, 6	bitr4i 266	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
8	7	necon3abii 2828	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
9		df-ex 1696	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
10	8, 9	bitr4i 266	1 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Ⅎwnfc 2738   ≠ wne 2780  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-nul 3875

This theorem is referenced by: (None)