Theorem inttsk 9596

Description: The intersection of a collection of Tarski classes is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
inttsk	|- ( ( A C_ Tarski /\ A =/= (/) ) -> |^| A e. Tarski )

Proof of Theorem inttsk

Dummy variables t z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 790	. . . . . . . 8 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) -> A C_ Tarski )
2	1	sselda 3603	. . . . . . 7 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) /\ t e. A ) -> t e. Tarski )
3		elinti 4485	. . . . . . . . 9 |- ( z e. |^| A -> ( t e. A -> z e. t ) )
4	3	imp 445	. . . . . . . 8 |- ( ( z e. |^| A /\ t e. A ) -> z e. t )
5	4	adantll 750	. . . . . . 7 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) /\ t e. A ) -> z e. t )
6		tskpwss 9574	. . . . . . 7 |- ( ( t e. Tarski /\ z e. t ) -> ~P z C_ t )
7	2, 5, 6	syl2anc 693	. . . . . 6 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) /\ t e. A ) -> ~P z C_ t )
8	7	ralrimiva 2966	. . . . 5 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) -> A. t e. A ~P z C_ t )
9		ssint 4493	. . . . 5 |- ( ~P z C_ |^| A <-> A. t e. A ~P z C_ t )
10	8, 9	sylibr 224	. . . 4 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) -> ~P z C_ |^| A )
11		tskpw 9575	. . . . . . 7 |- ( ( t e. Tarski /\ z e. t ) -> ~P z e. t )
12	2, 5, 11	syl2anc 693	. . . . . 6 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) /\ t e. A ) -> ~P z e. t )
13	12	ralrimiva 2966	. . . . 5 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) -> A. t e. A ~P z e. t )
14		vpwex 4849	. . . . . 6 |- ~P z e. _V
15	14	elint2 4482	. . . . 5 |- ( ~P z e. |^| A <-> A. t e. A ~P z e. t )
16	13, 15	sylibr 224	. . . 4 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) -> ~P z e. |^| A )
17	10, 16	jca 554	. . 3 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. |^| A ) -> ( ~P z C_ |^| A /\ ~P z e. |^| A ) )
18	17	ralrimiva 2966	. 2 |- ( ( A C_ Tarski /\ A =/= (/) ) -> A. z e. |^| A ( ~P z C_ |^| A /\ ~P z e. |^| A ) )
19		elpwi 4168	. . . 4 |- ( z e. ~P |^| A -> z C_ |^| A )
20		rexnal 2995	. . . . . . . 8 |- ( E. t e. A -. z e. t <-> -. A. t e. A z e. t )
21		simpr 477	. . . . . . . . . . . . 13 |- ( ( A C_ Tarski /\ A =/= (/) ) -> A =/= (/) )
22		intex 4820	. . . . . . . . . . . . 13 |- ( A =/= (/) <-> |^| A e. _V )
23	21, 22	sylib 208	. . . . . . . . . . . 12 |- ( ( A C_ Tarski /\ A =/= (/) ) -> |^| A e. _V )
24	23	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> |^| A e. _V )
25		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> z C_ |^| A )
26		ssdomg 8001	. . . . . . . . . . 11 |- ( |^| A e. _V -> ( z C_ |^| A -> z ~<_ |^| A ) )
27	24, 25, 26	sylc 65	. . . . . . . . . 10 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> z ~<_ |^| A )
28		vex 3203	. . . . . . . . . . . 12 |- t e. _V
29		intss1 4492	. . . . . . . . . . . . 13 |- ( t e. A -> |^| A C_ t )
30	29	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> |^| A C_ t )
31		ssdomg 8001	. . . . . . . . . . . 12 |- ( t e. _V -> ( |^| A C_ t -> |^| A ~<_ t ) )
32	28, 30, 31	mpsyl 68	. . . . . . . . . . 11 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> |^| A ~<_ t )
33		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> -. z e. t )
34		simplll 798	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> A C_ Tarski )
35		simprl 794	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> t e. A )
36	34, 35	sseldd 3604	. . . . . . . . . . . . . . 15 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> t e. Tarski )
37	25, 30	sstrd 3613	. . . . . . . . . . . . . . 15 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> z C_ t )
38		tsken 9576	. . . . . . . . . . . . . . 15 |- ( ( t e. Tarski /\ z C_ t ) -> ( z ~~ t \/ z e. t ) )
39	36, 37, 38	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> ( z ~~ t \/ z e. t ) )
40	39	ord 392	. . . . . . . . . . . . 13 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> ( -. z ~~ t -> z e. t ) )
41	33, 40	mt3d 140	. . . . . . . . . . . 12 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> z ~~ t )
42	41	ensymd 8007	. . . . . . . . . . 11 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> t ~~ z )
43		domentr 8015	. . . . . . . . . . 11 |- ( ( |^| A ~<_ t /\ t ~~ z ) -> |^| A ~<_ z )
44	32, 42, 43	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> |^| A ~<_ z )
45		sbth 8080	. . . . . . . . . 10 |- ( ( z ~<_ |^| A /\ |^| A ~<_ z ) -> z ~~ |^| A )
46	27, 44, 45	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) /\ ( t e. A /\ -. z e. t ) ) -> z ~~ |^| A )
47	46	rexlimdvaa 3032	. . . . . . . 8 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) -> ( E. t e. A -. z e. t -> z ~~ |^| A ) )
48	20, 47	syl5bir 233	. . . . . . 7 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) -> ( -. A. t e. A z e. t -> z ~~ |^| A ) )
49	48	con1d 139	. . . . . 6 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) -> ( -. z ~~ |^| A -> A. t e. A z e. t ) )
50		vex 3203	. . . . . . 7 |- z e. _V
51	50	elint2 4482	. . . . . 6 |- ( z e. |^| A <-> A. t e. A z e. t )
52	49, 51	syl6ibr 242	. . . . 5 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) -> ( -. z ~~ |^| A -> z e. |^| A ) )
53	52	orrd 393	. . . 4 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z C_ |^| A ) -> ( z ~~ |^| A \/ z e. |^| A ) )
54	19, 53	sylan2 491	. . 3 |- ( ( ( A C_ Tarski /\ A =/= (/) ) /\ z e. ~P |^| A ) -> ( z ~~ |^| A \/ z e. |^| A ) )
55	54	ralrimiva 2966	. 2 |- ( ( A C_ Tarski /\ A =/= (/) ) -> A. z e. ~P |^| A ( z ~~ |^| A \/ z e. |^| A ) )
56		eltsk2g 9573	. . 3 |- ( |^| A e. _V -> ( |^| A e. Tarski <-> ( A. z e. |^| A ( ~P z C_ |^| A /\ ~P z e. |^| A ) /\ A. z e. ~P |^| A ( z ~~ |^| A \/ z e. |^| A ) ) ) )
57	23, 56	syl 17	. 2 |- ( ( A C_ Tarski /\ A =/= (/) ) -> ( |^| A e. Tarski <-> ( A. z e. |^| A ( ~P z C_ |^| A /\ ~P z e. |^| A ) /\ A. z e. ~P |^| A ( z ~~ |^| A \/ z e. |^| A ) ) ) )
58	18, 55, 57	mpbir2and 957	1 |- ( ( A C_ Tarski /\ A =/= (/) ) -> |^| A e. Tarski )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   class class class wbr 4653   ~~ cen 7952   ~<_ cdom 7953   Tarskictsk 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-er 7742  df-en 7956  df-dom 7957  df-tsk 9571

This theorem is referenced by:  tskmcl  9663