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Theorem dfint3 31701
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 4442 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
2 ralnex 2986 . . . 4 (∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥 ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
3 vex 3189 . . . . . . . . 9 𝑦 ∈ V
4 vex 3189 . . . . . . . . 9 𝑥 ∈ V
53, 4brcnv 5265 . . . . . . . 8 (𝑦(V ∖ E )𝑥𝑥(V ∖ E )𝑦)
6 brv 31626 . . . . . . . . 9 𝑥V𝑦
7 brdif 4665 . . . . . . . . 9 (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦))
86, 7mpbiran 952 . . . . . . . 8 (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦)
95, 8bitr2i 265 . . . . . . 7 𝑥 E 𝑦𝑦(V ∖ E )𝑥)
109con1bii 346 . . . . . 6 𝑦(V ∖ E )𝑥𝑥 E 𝑦)
11 epel 4988 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
1210, 11bitr2i 265 . . . . 5 (𝑥𝑦 ↔ ¬ 𝑦(V ∖ E )𝑥)
1312ralbii 2974 . . . 4 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥)
14 eldif 3565 . . . . . 6 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴)))
154, 14mpbiran 952 . . . . 5 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴))
164elima 5430 . . . . 5 (𝑥 ∈ ((V ∖ E ) “ 𝐴) ↔ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
1715, 16xchbinx 324 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
182, 13, 173bitr4ri 293 . . 3 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ∀𝑦𝐴 𝑥𝑦)
1918abbi2i 2735 . 2 (V ∖ ((V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
201, 19eqtr4i 2646 1 𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  cdif 3552   cint 4440   class class class wbr 4613   E cep 4983  ccnv 5073  cima 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-int 4441  df-br 4614  df-opab 4674  df-eprel 4985  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by: (None)
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