Theorem dfint3 36342

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 4918	. 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
2		ralnex 3097	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥)
3		vex 3467	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4		vex 3467	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5869	. . . . . . . 8 ⊢ (𝑦◡(V ∖ E )𝑥 ↔ 𝑥(V ∖ E )𝑦)
6		brv 5455	. . . . . . . . 9 ⊢ 𝑥V𝑦
7		brdif 5168	. . . . . . . . 9 ⊢ (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦))
8	6, 7	mpbiran 721	. . . . . . . 8 ⊢ (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦)
9	5, 8	bitr2i 279	. . . . . . 7 ⊢ (¬ 𝑥 E 𝑦 ↔ 𝑦◡(V ∖ E )𝑥)
10	9	con1bii 359	. . . . . 6 ⊢ (¬ 𝑦◡(V ∖ E )𝑥 ↔ 𝑥 E 𝑦)
11		epel 5565	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
12	10, 11	bitr2i 279	. . . . 5 ⊢ (𝑥 ∈ 𝑦 ↔ ¬ 𝑦◡(V ∖ E )𝑥)
13	12	ralbii 3117	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥)
14		eldif 3923	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴)))
15	4, 14	mpbiran 721	. . . . 5 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴))
16	4	elima 6068	. . . . 5 ⊢ (𝑥 ∈ (◡(V ∖ E ) “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥)
17	15, 16	xchbinx 337	. . . 4 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥)
18	2, 13, 17	3bitr4ri 307	. . 3 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
19	18	eqabi 2904	. 2 ⊢ (V ∖ (◡(V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
20	1, 19	eqtr4i 2795	1 ⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴))

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910  ∩ cint 4916   class class class wbr 5113   E cep 5561  ^◡ccnv 5661   “ cima 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-int 4917  df-br 5114  df-opab 5178  df-eprel 5562  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by: (None)