Theorem ltrelxr 7959

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 7938	. 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 970	. . . . . 6 |- ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( ( x e. RR /\ y e. RR ) /\ x <RR y ) )
3	2	opabbii 4049	. . . . 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 4678	. . . . 5 |- { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ x <RR y ) } C_ ( RR X. RR )
5	3, 4	eqsstri 3174	. . . 4 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR X. RR )
6		rexpssxrxp 7943	. . . 4 |- ( RR X. RR ) C_ ( RR* X. RR* )
7	5, 6	sstri 3151	. . 3 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR* X. RR* )
8		ressxr 7942	. . . . . 6 |- RR C_ RR*
9		snsspr2 3722	. . . . . . 7 |- { -oo } C_ { +oo , -oo }
10		ssun2 3286	. . . . . . . 8 |- { +oo , -oo } C_ ( RR u. { +oo , -oo } )
11		df-xr 7937	. . . . . . . 8 |- RR* = ( RR u. { +oo , -oo } )
12	10, 11	sseqtrri 3177	. . . . . . 7 |- { +oo , -oo } C_ RR*
13	9, 12	sstri 3151	. . . . . 6 |- { -oo } C_ RR*
14	8, 13	unssi 3297	. . . . 5 |- ( RR u. { -oo } ) C_ RR*
15		snsspr1 3721	. . . . . 6 |- { +oo } C_ { +oo , -oo }
16	15, 12	sstri 3151	. . . . 5 |- { +oo } C_ RR*
17		xpss12 4711	. . . . 5 |- ( ( ( RR u. { -oo } ) C_ RR* /\ { +oo } C_ RR* ) -> ( ( RR u. { -oo } ) X. { +oo } ) C_ ( RR* X. RR* ) )
18	14, 16, 17	mp2an 423	. . . 4 |- ( ( RR u. { -oo } ) X. { +oo } ) C_ ( RR* X. RR* )
19		xpss12 4711	. . . . 5 |- ( ( { -oo } C_ RR* /\ RR C_ RR* ) -> ( { -oo } X. RR ) C_ ( RR* X. RR* ) )
20	13, 8, 19	mp2an 423	. . . 4 |- ( { -oo } X. RR ) C_ ( RR* X. RR* )
21	18, 20	unssi 3297	. . 3 |- ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) C_ ( RR* X. RR* )
22	7, 21	unssi 3297	. 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 3174	1 |- < C_ ( RR* X. RR* )

Syntax hints:   /\ wa 103   /\ w3a 968   e. wcel 2136   u. cun 3114   C_ wss 3116   {csn 3576   {cpr 3577   class class class wbr 3982   {copab 4042   X. cxp 4602   RRcr 7752   <RR cltrr 7757   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pr 3583  df-opab 4044  df-xp 4610  df-xr 7937  df-ltxr 7938

This theorem is referenced by:  ltrel  7960