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Theorem bj-restreg 33384
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 8656 . . 3 ((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)
2 eqcom 2778 . . . 4 ((𝑥𝐴) = ∅ ↔ ∅ = (𝑥𝐴))
32rexbii 3189 . . 3 (∃𝑥𝐴 (𝑥𝐴) = ∅ ↔ ∃𝑥𝐴 ∅ = (𝑥𝐴))
41, 3sylib 208 . 2 ((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 ∅ = (𝑥𝐴))
5 simpl 468 . . 3 ((𝐴𝑉𝐴 ≠ ∅) → 𝐴𝑉)
6 elrest 16296 . . 3 ((𝐴𝑉𝐴𝑉) → (∅ ∈ (𝐴t 𝐴) ↔ ∃𝑥𝐴 ∅ = (𝑥𝐴)))
75, 6syldan 579 . 2 ((𝐴𝑉𝐴 ≠ ∅) → (∅ ∈ (𝐴t 𝐴) ↔ ∃𝑥𝐴 ∅ = (𝑥𝐴)))
84, 7mpbird 247 1 ((𝐴𝑉𝐴 ≠ ∅) → ∅ ∈ (𝐴t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wrex 3062  cin 3722  c0 4063  (class class class)co 6793  t crest 16289
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-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096  ax-reg 8653
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-rest 16291
This theorem is referenced by: (None)
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