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Theorem dfint3 35953
Description: Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
Assertion
Ref Expression
dfint3 𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))

Proof of Theorem dfint3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfint2 4948 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
2 ralnex 3072 . . . 4 (∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥 ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
3 vex 3484 . . . . . . . . 9 𝑦 ∈ V
4 vex 3484 . . . . . . . . 9 𝑥 ∈ V
53, 4brcnv 5893 . . . . . . . 8 (𝑦(V ∖ E )𝑥𝑥(V ∖ E )𝑦)
6 brv 5477 . . . . . . . . 9 𝑥V𝑦
7 brdif 5196 . . . . . . . . 9 (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦))
86, 7mpbiran 709 . . . . . . . 8 (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦)
95, 8bitr2i 276 . . . . . . 7 𝑥 E 𝑦𝑦(V ∖ E )𝑥)
109con1bii 356 . . . . . 6 𝑦(V ∖ E )𝑥𝑥 E 𝑦)
11 epel 5587 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
1210, 11bitr2i 276 . . . . 5 (𝑥𝑦 ↔ ¬ 𝑦(V ∖ E )𝑥)
1312ralbii 3093 . . . 4 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥)
14 eldif 3961 . . . . . 6 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴)))
154, 14mpbiran 709 . . . . 5 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴))
164elima 6083 . . . . 5 (𝑥 ∈ ((V ∖ E ) “ 𝐴) ↔ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
1715, 16xchbinx 334 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
182, 13, 173bitr4ri 304 . . 3 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ∀𝑦𝐴 𝑥𝑦)
1918eqabi 2877 . 2 (V ∖ ((V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
201, 19eqtr4i 2768 1 𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  cdif 3948   cint 4946   class class class wbr 5143   E cep 5583  ccnv 5684  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-int 4947  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by: (None)
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