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Theorem dfint3 33297
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 4875 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
2 ralnex 3240 . . . 4 (∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥 ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
3 vex 3502 . . . . . . . . 9 𝑦 ∈ V
4 vex 3502 . . . . . . . . 9 𝑥 ∈ V
53, 4brcnv 5751 . . . . . . . 8 (𝑦(V ∖ E )𝑥𝑥(V ∖ E )𝑦)
6 brv 5360 . . . . . . . . 9 𝑥V𝑦
7 brdif 5115 . . . . . . . . 9 (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦))
86, 7mpbiran 705 . . . . . . . 8 (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦)
95, 8bitr2i 277 . . . . . . 7 𝑥 E 𝑦𝑦(V ∖ E )𝑥)
109con1bii 358 . . . . . 6 𝑦(V ∖ E )𝑥𝑥 E 𝑦)
11 epel 5467 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
1210, 11bitr2i 277 . . . . 5 (𝑥𝑦 ↔ ¬ 𝑦(V ∖ E )𝑥)
1312ralbii 3169 . . . 4 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥)
14 eldif 3949 . . . . . 6 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴)))
154, 14mpbiran 705 . . . . 5 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴))
164elima 5931 . . . . 5 (𝑥 ∈ ((V ∖ E ) “ 𝐴) ↔ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
1715, 16xchbinx 335 . . . 4 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
182, 13, 173bitr4ri 305 . . 3 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ∀𝑦𝐴 𝑥𝑦)
1918abbi2i 2957 . 2 (V ∖ ((V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
201, 19eqtr4i 2851 1 𝐴 = (V ∖ ((V ∖ E ) “ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  {cab 2803  wral 3142  wrex 3143  Vcvv 3499  cdif 3936   cint 4873   class class class wbr 5062   E cep 5462  ccnv 5552  cima 5556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-int 4874  df-br 5063  df-opab 5125  df-eprel 5463  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566
This theorem is referenced by: (None)
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