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Theorem zfreg 9105
 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 9220). (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 4247 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 219 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32anim2i 619 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (𝐴𝑉 ∧ ∃𝑥 𝑥𝐴))
4 zfregcl 9104 . . 3 (𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))
54imp 410 . 2 ((𝐴𝑉 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
6 disj 4347 . . . 4 ((𝑥𝐴) = ∅ ↔ ∀𝑦𝑥 ¬ 𝑦𝐴)
76rexbii 3175 . . 3 (∃𝑥𝐴 (𝑥𝐴) = ∅ ↔ ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)
87biimpri 231 . 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 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071   ∩ cin 3859  ∅c0 4227 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729  ax-reg 9102 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-dif 3863  df-in 3867  df-nul 4228 This theorem is referenced by:  zfregfr  9114  en3lp  9123  inf3lem3  9139  bj-restreg  34829  setindtr  40383
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