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Theorem zfreg2OLD 8463
Description: Alternate version of zfreg 8460 obsolete as of 28-Apr-2021. (Contributed by NM, 17-Sep-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
zfreg2OLD.1 𝐴 ∈ V
Assertion
Ref Expression
zfreg2OLD (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝐴𝑥) = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem zfreg2OLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 zfreg2OLD.1 . . 3 𝐴 ∈ V
21zfregclOLD 8461 . 2 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
3 n0 3913 . 2 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
4 disjr 3996 . . 3 ((𝐴𝑥) = ∅ ↔ ∀𝑦𝑥 ¬ 𝑦𝐴)
54rexbii 3036 . 2 (∃𝑥𝐴 (𝐴𝑥) = ∅ ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
62, 3, 53imtr4i 281 1 (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝐴𝑥) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2908  wrex 2909  Vcvv 3190  cin 3559  c0 3897
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  ax-reg 8457
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-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-in 3567  df-nul 3898
This theorem is referenced by: (None)
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