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Theorem zfreg 8656
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 8772). (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 4078 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 206 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim2i 603 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (𝐴𝑉 ∧ ∃𝑥 𝑥𝐴))
4 zfregcl 8655 . . 3 (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
54imp 393 . 2 ((𝐴𝑉 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
6 disj 4160 . . . 4 ((𝑥𝐴) = ∅ ↔ ∀𝑦𝑥 ¬ 𝑦𝐴)
76rexbii 3189 . . 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 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  cin 3722  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-reg 8653
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064
This theorem is referenced by:  zfregfr  8665  en3lp  8673  inf3lem3  8691  bj-restreg  33384  setindtr  38117
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