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 "kA)

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 e. _V
2	1	eluni1 4174	. . . . . 6 |- (x e. ⋃1(`'k ∼ Sk "kA) <-> {x} e. (`'k ∼ Sk "kA))
3		snex 4112	. . . . . . 7 |- {x} e. _V
4	3	elimak 4260	. . . . . 6 |- ({x} e. (`'k ∼ Sk "kA) <-> E.y e. A <<y, {x}>> e. `'k ∼ Sk )
5	2, 4	bitri 240	. . . . 5 |- (x e. ⋃1(`'k ∼ Sk "kA) <-> E.y e. A <<y, {x}>> e. `'k ∼ Sk )
6		vex 2863	. . . . . . . 8 |- y e. _V
7	6, 3	opkelcnvk 4251	. . . . . . 7 |- (<<y, {x}>> e. `'k ∼ Sk <-> <<{x}, y>> e. ∼ Sk )
8		opkex 4114	. . . . . . . 8 |- <<{x}, y>> e. _V
9	8	elcompl 3226	. . . . . . 7 |- (<<{x}, y>> e. ∼ Sk <-> -. <<{x}, y>> e. Sk )
10	1, 6	elssetk 4271	. . . . . . . 8 |- (<<{x}, y>> e. Sk <-> x e. y)
11	10	notbii 287	. . . . . . 7 |- (-. <<{x}, y>> e. Sk <-> -. x e. y)
12	7, 9, 11	3bitri 262	. . . . . 6 |- (<<y, {x}>> e. `'k ∼ Sk <-> -. x e. y)
13	12	rexbii 2640	. . . . 5 |- (E.y e. A <<y, {x}>> e. `'k ∼ Sk <-> E.y e. A -. x e. y)
14		rexnal 2626	. . . . 5 |- (E.y e. A -. x e. y <-> -. A.y e. A x e. y)
15	5, 13, 14	3bitri 262	. . . 4 |- (x e. ⋃1(`'k ∼ Sk "kA) <-> -. A.y e. A x e. y)
16	15	con2bii 322	. . 3 |- (A.y e. A x e. y <-> -. x e. ⋃1(`'k ∼ Sk "kA))
17	1	elint2 3934	. . 3 |- (x e. |^|A <-> A.y e. A x e. y)
18	1	elcompl 3226	. . 3 |- (x e. ∼ ⋃1(`'k ∼ Sk "kA) <-> -. x e. ⋃1(`'k ∼ Sk "kA))
19	16, 17, 18	3bitr4i 268	. 2 |- (x e. |^|A <-> x e. ∼ ⋃1(`'k ∼ Sk "kA))
20	19	eqriv 2350	1 |- |^|A = ∼ ⋃1(`'k ∼ Sk "kA)

Syntax hints:  -. wn 3   = wceq 1642   e. wcel 1710  A.wral 2615  E.wrex 2616   ∼ ccompl 3206  {csn 3738  |^|cint 3927  <<copk 4058  ⋃₁cuni1 4134  `'_kccnvk 4176  "_kcimak 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