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Theorem n0fOLD 3904
 Description: Obsolete proof of n0f 3903 as of 15-Jul-2021. (Contributed by NM, 17-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
eq0f.1 𝑥𝐴
Assertion
Ref Expression
n0fOLD (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)

Proof of Theorem n0fOLD
StepHypRef Expression
1 eq0f.1 . . . . 5 𝑥𝐴
2 nfcv 2761 . . . . 5 𝑥
31, 2cleqf 2786 . . . 4 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
4 noel 3895 . . . . . 6 ¬ 𝑥 ∈ ∅
54nbn 362 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
65albii 1744 . . . 4 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
73, 6bitr4i 267 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
87necon3abii 2836 . 2 (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥𝐴)
9 df-ex 1702 . 2 (∃𝑥 𝑥𝐴 ↔ ¬ ∀𝑥 ¬ 𝑥𝐴)
108, 9bitr4i 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   ≠ wne 2790  ∅c0 3891 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-ne 2791  df-v 3188  df-dif 3558  df-nul 3892 This theorem is referenced by: (None)
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