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Theorem zfreg 8444
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 8552). (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 3907 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 206 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim2i 592 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (𝐴𝑉 ∧ ∃𝑥 𝑥𝐴))
4 zfregcl 8443 . . 3 (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
54imp 445 . 2 ((𝐴𝑉 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
6 disj 3989 . . . 4 ((𝑥𝐴) = ∅ ↔ ∀𝑦𝑥 ¬ 𝑦𝐴)
76rexbii 3034 . . 3 (∃𝑥𝐴 (𝑥𝐴) = ∅ ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
87biimpri 218 . 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 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  cin 3554  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  ax-reg 8441
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 2912  df-rex 2913  df-v 3188  df-dif 3558  df-in 3562  df-nul 3892
This theorem is referenced by:  zfregfr  8453  en3lp  8457  inf3lem3  8471  bj-restreg  32686  setindtr  37068
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