Theorem mreincl 16259

Description: Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
mreincl	|- ( ( C e. (Moore ` X ) /\ A e. C /\ B e. C ) -> ( A i^i B ) e. C )

Proof of Theorem mreincl

Step	Hyp	Ref	Expression
1		intprg 4511	. . 3 |- ( ( A e. C /\ B e. C ) -> |^| { A , B } = ( A i^i B ) )
2	1	3adant1 1079	. 2 |- ( ( C e. (Moore ` X ) /\ A e. C /\ B e. C ) -> |^| { A , B } = ( A i^i B ) )
3		simp1 1061	. . 3 |- ( ( C e. (Moore ` X ) /\ A e. C /\ B e. C ) -> C e. (Moore ` X ) )
4		prssi 4353	. . . 4 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
5	4	3adant1 1079	. . 3 |- ( ( C e. (Moore ` X ) /\ A e. C /\ B e. C ) -> { A , B } C_ C )
6		prnzg 4311	. . . 4 |- ( A e. C -> { A , B } =/= (/) )
7	6	3ad2ant2 1083	. . 3 |- ( ( C e. (Moore ` X ) /\ A e. C /\ B e. C ) -> { A , B } =/= (/) )
8		mreintcl 16255	. . 3 |- ( ( C e. (Moore ` X ) /\ { A , B } C_ C /\ { A , B } =/= (/) ) -> |^| { A , B } e. C )
9	3, 5, 7, 8	syl3anc 1326	. 2 |- ( ( C e. (Moore ` X ) /\ A e. C /\ B e. C ) -> |^| { A , B } e. C )
10	2, 9	eqeltrrd 2702	1 |- ( ( C e. (Moore ` X ) /\ A e. C /\ B e. C ) -> ( A i^i B ) e. C )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   {cpr 4179   |^|cint 4475   ` cfv 5888  Moorecmre 16242

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

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-sbc 3436  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-mre 16246

This theorem is referenced by:  submacs  17365  subgacs  17629  nsgacs  17630  lsmmod  18088  lssacs  18967  mreclatdemoBAD  20900  subrgacs  37770  sdrgacs  37771