Theorem dfint3 4319

Description: Alternate definition of class intersection for the existence proof. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
dfint3	⊢ ∩A = ∼ ⋃1(◡k ∼ Sk “k A)

Proof of Theorem dfint3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . 7 ⊢ x ∈ V
2	1	eluni1 4174	. . . . . 6 ⊢ (x ∈ ⋃1(◡k ∼ Sk “k A) ↔ {x} ∈ (◡k ∼ Sk “k A))
3		snex 4112	. . . . . . 7 ⊢ {x} ∈ V
4	3	elimak 4260	. . . . . 6 ⊢ ({x} ∈ (◡k ∼ Sk “k A) ↔ ∃y ∈ A ⟪y, {x}⟫ ∈ ◡k ∼ Sk )
5	2, 4	bitri 240	. . . . 5 ⊢ (x ∈ ⋃1(◡k ∼ Sk “k A) ↔ ∃y ∈ A ⟪y, {x}⟫ ∈ ◡k ∼ Sk )
6		vex 2863	. . . . . . . 8 ⊢ y ∈ V
7	6, 3	opkelcnvk 4251	. . . . . . 7 ⊢ (⟪y, {x}⟫ ∈ ◡k ∼ Sk ↔ ⟪{x}, y⟫ ∈ ∼ Sk )
8		opkex 4114	. . . . . . . 8 ⊢ ⟪{x}, y⟫ ∈ V
9	8	elcompl 3226	. . . . . . 7 ⊢ (⟪{x}, y⟫ ∈ ∼ Sk ↔ ¬ ⟪{x}, y⟫ ∈ Sk )
10	1, 6	elssetk 4271	. . . . . . . 8 ⊢ (⟪{x}, y⟫ ∈ Sk ↔ x ∈ y)
11	10	notbii 287	. . . . . . 7 ⊢ (¬ ⟪{x}, y⟫ ∈ Sk ↔ ¬ x ∈ y)
12	7, 9, 11	3bitri 262	. . . . . 6 ⊢ (⟪y, {x}⟫ ∈ ◡k ∼ Sk ↔ ¬ x ∈ y)
13	12	rexbii 2640	. . . . 5 ⊢ (∃y ∈ A ⟪y, {x}⟫ ∈ ◡k ∼ Sk ↔ ∃y ∈ A ¬ x ∈ y)
14		rexnal 2626	. . . . 5 ⊢ (∃y ∈ A ¬ x ∈ y ↔ ¬ ∀y ∈ A x ∈ y)
15	5, 13, 14	3bitri 262	. . . 4 ⊢ (x ∈ ⋃1(◡k ∼ Sk “k A) ↔ ¬ ∀y ∈ A x ∈ y)
16	15	con2bii 322	. . 3 ⊢ (∀y ∈ A x ∈ y ↔ ¬ x ∈ ⋃1(◡k ∼ Sk “k A))
17	1	elint2 3934	. . 3 ⊢ (x ∈ ∩A ↔ ∀y ∈ A x ∈ y)
18	1	elcompl 3226	. . 3 ⊢ (x ∈ ∼ ⋃1(◡k ∼ Sk “k A) ↔ ¬ x ∈ ⋃1(◡k ∼ Sk “k A))
19	16, 17, 18	3bitr4i 268	. 2 ⊢ (x ∈ ∩A ↔ x ∈ ∼ ⋃1(◡k ∼ Sk “k A))
20	19	eqriv 2350	1 ⊢ ∩A = ∼ ⋃1(◡k ∼ Sk “k A)

Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616   ∼ ccompl 3206  {csn 3738  ∩cint 3927  ⟪copk 4058  ⋃₁cuni1 4134  ^◡_kccnvk 4176   “_k cimak 4180   S_k cssetk 4184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-uni1 4139  df-cnvk 4187  df-imak 4190  df-ssetk 4194

This theorem is referenced by:  intexg  4320