Theorem n0f 3559

Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3560 requires only that x not be free in, rather than not occur in, A. (Contributed by NM, 17-Oct-2003.)

Hypothesis

Ref	Expression
n0f.1	⊢ ℲxA

Assertion

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

Proof of Theorem n0f

Step	Hyp	Ref	Expression
1		n0f.1	. . . . 5 ⊢ ℲxA
2		nfcv 2490	. . . . 5 ⊢ Ⅎx∅
3	1, 2	cleqf 2514	. . . 4 ⊢ (A = ∅ ↔ ∀x(x ∈ A ↔ x ∈ ∅))
4		noel 3555	. . . . . 6 ⊢ ¬ x ∈ ∅
5	4	nbn 336	. . . . 5 ⊢ (¬ x ∈ A ↔ (x ∈ A ↔ x ∈ ∅))
6	5	albii 1566	. . . 4 ⊢ (∀x ¬ x ∈ A ↔ ∀x(x ∈ A ↔ x ∈ ∅))
7	3, 6	bitr4i 243	. . 3 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A)
8	7	necon3abii 2547	. 2 ⊢ (A ≠ ∅ ↔ ¬ ∀x ¬ x ∈ A)
9		df-ex 1542	. 2 ⊢ (∃x x ∈ A ↔ ¬ ∀x ¬ x ∈ A)
10	8, 9	bitr4i 243	1 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477   ≠ wne 2517  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  n0  3560  abn0  3569