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Theorem neq0f 3907
 Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of neq0 3911 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by BJ, 15-Jul-2021.)
Hypothesis
Ref Expression
eq0f.1 𝑥𝐴
Assertion
Ref Expression
neq0f 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)

Proof of Theorem neq0f
StepHypRef Expression
1 eq0f.1 . . . 4 𝑥𝐴
21eq0f 3906 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
32notbii 310 . 2 𝐴 = ∅ ↔ ¬ ∀𝑥 ¬ 𝑥𝐴)
4 df-ex 1702 . 2 (∃𝑥 𝑥𝐴 ↔ ¬ ∀𝑥 ¬ 𝑥𝐴)
53, 4bitr4i 267 1 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987  Ⅎwnfc 2748  ∅c0 3896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-dif 3562  df-nul 3897 This theorem is referenced by:  n0f  3908  neq0  3911
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