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Theorem zfreg 9051
Description: The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form". Axiom Reg of [BellMachover] p. 480. There is also a "strong form", not requiring that 𝐴 be a set, that can be proved with more difficulty (see zfregs 9166). (Contributed by NM, 26-Nov-1995.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.)
Assertion
Ref Expression
zfreg ((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem zfreg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 n0 4313 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 217 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim2i 616 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (𝐴𝑉 ∧ ∃𝑥 𝑥𝐴))
4 zfregcl 9050 . . 3 (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
54imp 407 . 2 ((𝐴𝑉 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
6 disj 4401 . . . 4 ((𝑥𝐴) = ∅ ↔ ∀𝑦𝑥 ¬ 𝑦𝐴)
76rexbii 3251 . . 3 (∃𝑥𝐴 (𝑥𝐴) = ∅ ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
87biimpri 229 . 2 (∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴 → ∃𝑥𝐴 (𝑥𝐴) = ∅)
93, 5, 83syl 18 1 ((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wex 1773  wcel 2107  wne 3020  wral 3142  wrex 3143  cin 3938  c0 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-reg 9048
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-dif 3942  df-in 3946  df-nul 4295
This theorem is referenced by:  zfregfr  9060  en3lp  9069  inf3lem3  9085  bj-restreg  34273  setindtr  39488
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