Theorem n0f 1713

Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 1714 requires only that x not be free in, rather than not occur in, A.

Hypothesis

Ref	Expression
nnullf.1	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
n0f	⊢ (¬ A = ∅ ↔ ∃x x ∈ A)

Distinct variable group(s):   x,y   y,A

Proof of Theorem n0f

Step	Hyp	Ref	Expression
1		nnullf.1	. . . . 5 ⊢ (y ∈ A → ∀x y ∈ A)
2		ax-17 925	. . . . 5 ⊢ (y ∈ ∅ → ∀x y ∈ ∅)
3	1, 2	cleqf 1167	. . . 4 ⊢ (A = ∅ ↔ ∀x(x ∈ A ↔ x ∈ ∅))
4		noel 1711	. . . . . 6 ⊢ ¬ x ∈ ∅
5	4	nbn 542	. . . . 5 ⊢ (¬ x ∈ A ↔ (x ∈ A ↔ x ∈ ∅))
6	5	bial 695	. . . 4 ⊢ (∀x ¬ x ∈ A ↔ ∀x(x ∈ A ↔ x ∈ ∅))
7	3, 6	bitr4 154	. . 3 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A)
8	7	negbii 162	. 2 ⊢ (¬ A = ∅ ↔ ¬ ∀x ¬ x ∈ A)
9		df-ex 679	. 2 ⊢ (∃x x ∈ A ↔ ¬ ∀x ¬ x ∈ A)
10	8, 9	bitr4 154	1 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∅c0 1707

This theorem is referenced by:  n0 1714  cp 3547

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-nul 1708

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