Theorem cvmsdisj 31252

Description: An even covering of U is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypothesis

Ref	Expression
cvmcov.1	|- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )

Assertion

Ref	Expression
cvmsdisj	|- ( ( T e. ( S ` U ) /\ A e. T /\ B e. T ) -> ( A = B \/ ( A i^i B ) = (/) ) )

Distinct variable groups:   k, s, u, v, C   k, F, s, u, v   k, J, s, u, v   U, k, s, u, v   T, s, u, v   u, A, v   v, B

Allowed substitution hints:   A( k, s)   B( u, k, s)   S( v, u, k, s)   T( k)

Proof of Theorem cvmsdisj

Step	Hyp	Ref	Expression
1		df-ne 2795	. . 3 |- ( A =/= B <-> -. A = B )
2		cvmcov.1	. . . . . . . . . . 11 |- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
3	2	cvmsi 31247	. . . . . . . . . 10 |- ( T e. ( S ` U ) -> ( U e. J /\ ( T C_ C /\ T =/= (/) ) /\ ( U. T = ( `' F " U ) /\ A. u e. T ( A. v e. ( T \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t U ) ) ) ) ) )
4	3	simp3d 1075	. . . . . . . . 9 |- ( T e. ( S ` U ) -> ( U. T = ( `' F " U ) /\ A. u e. T ( A. v e. ( T \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t U ) ) ) ) )
5	4	simprd 479	. . . . . . . 8 |- ( T e. ( S ` U ) -> A. u e. T ( A. v e. ( T \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t U ) ) ) )
6		simpl 473	. . . . . . . . 9 |- ( ( A. v e. ( T \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t U ) ) ) -> A. v e. ( T \ { u } ) ( u i^i v ) = (/) )
7	6	ralimi 2952	. . . . . . . 8 |- ( A. u e. T ( A. v e. ( T \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t U ) ) ) -> A. u e. T A. v e. ( T \ { u } ) ( u i^i v ) = (/) )
8	5, 7	syl 17	. . . . . . 7 |- ( T e. ( S ` U ) -> A. u e. T A. v e. ( T \ { u } ) ( u i^i v ) = (/) )
9		sneq 4187	. . . . . . . . . 10 |- ( u = A -> { u } = { A } )
10	9	difeq2d 3728	. . . . . . . . 9 |- ( u = A -> ( T \ { u } ) = ( T \ { A } ) )
11		ineq1 3807	. . . . . . . . . 10 |- ( u = A -> ( u i^i v ) = ( A i^i v ) )
12	11	eqeq1d 2624	. . . . . . . . 9 |- ( u = A -> ( ( u i^i v ) = (/) <-> ( A i^i v ) = (/) ) )
13	10, 12	raleqbidv 3152	. . . . . . . 8 |- ( u = A -> ( A. v e. ( T \ { u } ) ( u i^i v ) = (/) <-> A. v e. ( T \ { A } ) ( A i^i v ) = (/) ) )
14	13	rspccva 3308	. . . . . . 7 |- ( ( A. u e. T A. v e. ( T \ { u } ) ( u i^i v ) = (/) /\ A e. T ) -> A. v e. ( T \ { A } ) ( A i^i v ) = (/) )
15	8, 14	sylan 488	. . . . . 6 |- ( ( T e. ( S ` U ) /\ A e. T ) -> A. v e. ( T \ { A } ) ( A i^i v ) = (/) )
16		necom 2847	. . . . . . 7 |- ( A =/= B <-> B =/= A )
17		eldifsn 4317	. . . . . . . 8 |- ( B e. ( T \ { A } ) <-> ( B e. T /\ B =/= A ) )
18	17	biimpri 218	. . . . . . 7 |- ( ( B e. T /\ B =/= A ) -> B e. ( T \ { A } ) )
19	16, 18	sylan2b 492	. . . . . 6 |- ( ( B e. T /\ A =/= B ) -> B e. ( T \ { A } ) )
20		ineq2 3808	. . . . . . . 8 |- ( v = B -> ( A i^i v ) = ( A i^i B ) )
21	20	eqeq1d 2624	. . . . . . 7 |- ( v = B -> ( ( A i^i v ) = (/) <-> ( A i^i B ) = (/) ) )
22	21	rspccv 3306	. . . . . 6 |- ( A. v e. ( T \ { A } ) ( A i^i v ) = (/) -> ( B e. ( T \ { A } ) -> ( A i^i B ) = (/) ) )
23	15, 19, 22	syl2im 40	. . . . 5 |- ( ( T e. ( S ` U ) /\ A e. T ) -> ( ( B e. T /\ A =/= B ) -> ( A i^i B ) = (/) ) )
24	23	expd 452	. . . 4 |- ( ( T e. ( S ` U ) /\ A e. T ) -> ( B e. T -> ( A =/= B -> ( A i^i B ) = (/) ) ) )
25	24	3impia 1261	. . 3 |- ( ( T e. ( S ` U ) /\ A e. T /\ B e. T ) -> ( A =/= B -> ( A i^i B ) = (/) ) )
26	1, 25	syl5bir 233	. 2 |- ( ( T e. ( S ` U ) /\ A e. T /\ B e. T ) -> ( -. A = B -> ( A i^i B ) = (/) ) )
27	26	orrd 393	1 |- ( ( T e. ( S ` U ) /\ A e. T /\ B e. T ) -> ( A = B \/ ( A i^i B ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Homeochmeo 21556

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-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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  cvmscld  31255  cvmsss2  31256  cvmseu  31258