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Theorem zfregOLD 8385
 Description: Obsolete version of zfreg 8383 as of 28-Apr-2021. (Contributed by NM, 26-Nov-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
zfregOLD.1 𝐴 ∈ V
Assertion
Ref Expression
zfregOLD (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝑥𝐴) = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem zfregOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 zfregOLD.1 . . 3 𝐴 ∈ V
21zfregclOLD 8384 . 2 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
3 n0 3890 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
4 disj 3969 . . 3 ((𝑥𝐴) = ∅ ↔ ∀𝑦𝑥 ¬ 𝑦𝐴)
54rexbii 3023 . 2 (∃𝑥𝐴 (𝑥𝐴) = ∅ ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
62, 3, 53imtr4i 280 1 (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝑥𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-nul 3875 This theorem is referenced by: (None)
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