Theorem dfint3 36840

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 4908	. 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
2		ralnex 3088	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥)
3		vex 3454	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4		vex 3454	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5861	. . . . . . . 8 ⊢ (𝑦◡(V ∖ E )𝑥 ↔ 𝑥(V ∖ E )𝑦)
6		brv 5441	. . . . . . . . 9 ⊢ 𝑥V𝑦
7		brdif 5157	. . . . . . . . 9 ⊢ (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦))
8	6, 7	mpbiran 722	. . . . . . . 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 5554	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
12	10, 11	bitr2i 279	. . . . 5 ⊢ (𝑥 ∈ 𝑦 ↔ ¬ 𝑦◡(V ∖ E )𝑥)
13	12	ralbii 3108	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥)
14		eldif 4048	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴)))
15	4, 14	mpbiran 722	. . . . 5 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴))
16	4	elima 6063	. . . . 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 2895	. 2 ⊢ (V ∖ (◡(V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
20	1, 19	eqtr4i 2786	1 ⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∖ cdif 4044  ∩ cint 4906   class class class wbr 5102   E cep 5550  ^◡ccnv 5650   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-int 4907  df-br 5103  df-opab 5167  df-eprel 5551  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by: (None)