Theorem dfint3 36168

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.

Step	Hyp	Ref	Expression
1		dfint2 4906	. 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
2		ralnex 3064	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥)
3		vex 3446	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4		vex 3446	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5839	. . . . . . . 8 ⊢ (𝑦◡(V ∖ E )𝑥 ↔ 𝑥(V ∖ E )𝑦)
6		brv 5428	. . . . . . . . 9 ⊢ 𝑥V𝑦
7		brdif 5153	. . . . . . . . 9 ⊢ (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦))
8	6, 7	mpbiran 710	. . . . . . . 8 ⊢ (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦)
9	5, 8	bitr2i 276	. . . . . . 7 ⊢ (¬ 𝑥 E 𝑦 ↔ 𝑦◡(V ∖ E )𝑥)
10	9	con1bii 356	. . . . . 6 ⊢ (¬ 𝑦◡(V ∖ E )𝑥 ↔ 𝑥 E 𝑦)
11		epel 5535	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
12	10, 11	bitr2i 276	. . . . 5 ⊢ (𝑥 ∈ 𝑦 ↔ ¬ 𝑦◡(V ∖ E )𝑥)
13	12	ralbii 3084	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥)
14		eldif 3913	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴)))
15	4, 14	mpbiran 710	. . . . 5 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴))
16	4	elima 6032	. . . . 5 ⊢ (𝑥 ∈ (◡(V ∖ E ) “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥)
17	15, 16	xchbinx 334	. . . 4 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥)
18	2, 13, 17	3bitr4ri 304	. . 3 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
19	18	eqabi 2872	. 2 ⊢ (V ∖ (◡(V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
20	1, 19	eqtr4i 2763	1 ⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴))

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∖ cdif 3900  ∩ cint 4904   class class class wbr 5100   E cep 5531  ^◡ccnv 5631   “ cima 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-int 4905  df-br 5101  df-opab 5163  df-eprel 5532  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645

This theorem is referenced by: (None)