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Theorem bj-restreg 32233
 Description: A reformulation of the axiom of regularity using elementwise intersection. (RK: might have to be placed later since theorems in this section are to be moved early (in the section related to the algebra of sets).) (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-restreg ((𝐴𝑉𝐴 ≠ ∅) → ∅ ∈ (𝐴t 𝐴))

Proof of Theorem bj-restreg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 zfreg 8383 . . 3 ((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)
2 eqcom 2617 . . . 4 ((𝑥𝐴) = ∅ ↔ ∅ = (𝑥𝐴))
32rexbii 3023 . . 3 (∃𝑥𝐴 (𝑥𝐴) = ∅ ↔ ∃𝑥𝐴 ∅ = (𝑥𝐴))
41, 3sylib 207 . 2 ((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 ∅ = (𝑥𝐴))
5 simpl 472 . . 3 ((𝐴𝑉𝐴 ≠ ∅) → 𝐴𝑉)
6 elrest 15911 . . 3 ((𝐴𝑉𝐴𝑉) → (∅ ∈ (𝐴t 𝐴) ↔ ∃𝑥𝐴 ∅ = (𝑥𝐴)))
75, 6syldan 486 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (∅ ∈ (𝐴t 𝐴) ↔ ∃𝑥𝐴 ∅ = (𝑥𝐴)))
84, 7mpbird 246 1 ((𝐴𝑉𝐴 ≠ ∅) → ∅ ∈ (𝐴t 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∩ cin 3539  ∅c0 3874  (class class class)co 6549   ↾t crest 15904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rest 15906 This theorem is referenced by: (None)
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