MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  n0f Structured version   Visualization version   GIF version

Theorem n0f 3881
Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3885 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by NM, 17-Oct-2003.)
Hypothesis
Ref Expression
eq0f.1 𝑥𝐴
Assertion
Ref Expression
n0f (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)

Proof of Theorem n0f
StepHypRef Expression
1 df-ne 2777 . 2 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
2 eq0f.1 . . 3 𝑥𝐴
32neq0f 3880 . 2 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
41, 3bitri 262 1 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194   = wceq 1474  wex 1694  wcel 1975  wnfc 2733  wne 2775  c0 3869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-v 3170  df-dif 3538  df-nul 3870
This theorem is referenced by:  n0  3885  abn0  3903  cp  8610  ordtconlem1  29100  inn0f  38067
  Copyright terms: Public domain W3C validator