Theorem ltrelxr 7173

Description: 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
ltrelxr	|- < C_ ( RR* X. RR* )

Proof of Theorem ltrelxr

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

Step	Hyp	Ref	Expression
1		df-ltxr 7158	. 2 |- < = ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) )
2		df-3an 921	. . . . . 6 |- ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( ( x e. RR /\ y e. RR ) /\ x <RR y ) )
3	2	opabbii 3845	. . . . 5 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ x <RR y ) }
4		opabssxp 4432	. . . . 5 |- { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ x <RR y ) } C_ ( RR X. RR )
5	3, 4	eqsstri 3029	. . . 4 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR X. RR )
6		rexpssxrxp 7163	. . . 4 |- ( RR X. RR ) C_ ( RR* X. RR* )
7	5, 6	sstri 3008	. . 3 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR* X. RR* )
8		ressxr 7162	. . . . . 6 |- RR C_ RR*
9		snsspr2 3534	. . . . . . 7 |- { -oo } C_ { +oo , -oo }
10		ssun2 3136	. . . . . . . 8 |- { +oo , -oo } C_ ( RR u. { +oo , -oo } )
11		df-xr 7157	. . . . . . . 8 |- RR* = ( RR u. { +oo , -oo } )
12	10, 11	sseqtr4i 3032	. . . . . . 7 |- { +oo , -oo } C_ RR*
13	9, 12	sstri 3008	. . . . . 6 |- { -oo } C_ RR*
14	8, 13	unssi 3147	. . . . 5 |- ( RR u. { -oo } ) C_ RR*
15		snsspr1 3533	. . . . . 6 |- { +oo } C_ { +oo , -oo }
16	15, 12	sstri 3008	. . . . 5 |- { +oo } C_ RR*
17		xpss12 4463	. . . . 5 |- ( ( ( RR u. { -oo } ) C_ RR* /\ { +oo } C_ RR* ) -> ( ( RR u. { -oo } ) X. { +oo } ) C_ ( RR* X. RR* ) )
18	14, 16, 17	mp2an 416	. . . 4 |- ( ( RR u. { -oo } ) X. { +oo } ) C_ ( RR* X. RR* )
19		xpss12 4463	. . . . 5 |- ( ( { -oo } C_ RR* /\ RR C_ RR* ) -> ( { -oo } X. RR ) C_ ( RR* X. RR* ) )
20	13, 8, 19	mp2an 416	. . . 4 |- ( { -oo } X. RR ) C_ ( RR* X. RR* )
21	18, 20	unssi 3147	. . 3 |- ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) C_ ( RR* X. RR* )
22	7, 21	unssi 3147	. 2 |- ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) ) C_ ( RR* X. RR* )
23	1, 22	eqsstri 3029	1 |- < C_ ( RR* X. RR* )

Syntax hints:   /\ wa 102   /\ w3a 919   e. wcel 1433   u. cun 2971   C_ wss 2973   {csn 3398   {cpr 3399   class class class wbr 3785   {copab 3838   X. cxp 4361   RRcr 6980   <RR cltrr 6985   +oocpnf 7150   -oocmnf 7151   RR*cxr 7152   < clt 7153

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pr 3405  df-opab 3840  df-xp 4369  df-xr 7157  df-ltxr 7158

This theorem is referenced by:  ltrel  7174