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Theorem dfint3 32043
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 4475 . 2 𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
2 ralnex 2991 . . . 4 (∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥 ↔ ¬ ∃𝑦𝐴 𝑦(V ∖ E )𝑥)
3 vex 3201 . . . . . . . . 9 𝑦 ∈ V
4 vex 3201 . . . . . . . . 9 𝑥 ∈ V
53, 4brcnv 5303 . . . . . . . 8 (𝑦(V ∖ E )𝑥𝑥(V ∖ E )𝑦)
6 brv 4939 . . . . . . . . 9 𝑥V𝑦
7 brdif 4703 . . . . . . . . 9 (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦))
86, 7mpbiran 953 . . . . . . . 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 5030 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
1210, 11bitr2i 265 . . . . 5 (𝑥𝑦 ↔ ¬ 𝑦(V ∖ E )𝑥)
1312ralbii 2979 . . . 4 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦𝐴 ¬ 𝑦(V ∖ E )𝑥)
14 eldif 3582 . . . . . 6 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴)))
154, 14mpbiran 953 . . . . 5 (𝑥 ∈ (V ∖ ((V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ ((V ∖ E ) “ 𝐴))
164elima 5469 . . . . 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 2737 . 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 1482  wcel 1989  {cab 2607  wral 2911  wrex 2912  Vcvv 3198  cdif 3569   cint 4473   class class class wbr 4651   E cep 5026  ccnv 5111  cima 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-int 4474  df-br 4652  df-opab 4711  df-eprel 5027  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125
This theorem is referenced by: (None)
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