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Theorem zfreg 8743
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 8859). (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 4132 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 208 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim2i 611 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (𝐴𝑉 ∧ ∃𝑥 𝑥𝐴))
4 zfregcl 8742 . . 3 (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
54imp 396 . 2 ((𝐴𝑉 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
6 disj 4213 . . . 4 ((𝑥𝐴) = ∅ ↔ ∀𝑦𝑥 ¬ 𝑦𝐴)
76rexbii 3223 . . 3 (∃𝑥𝐴 (𝑥𝐴) = ∅ ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
87biimpri 220 . 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 385   = wceq 1653  wex 1875  wcel 2157  wne 2972  wral 3090  wrex 3091  cin 3769  c0 4116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-reg 8740
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-v 3388  df-dif 3773  df-in 3777  df-nul 4117
This theorem is referenced by:  zfregfr  8752  en3lp  8760  inf3lem3  8778  bj-restreg  33544  setindtr  38371
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