Theorem ltrelxr 7818

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 7798	. 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 964	. . . . . 6 |- ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( ( x e. RR /\ y e. RR ) /\ x <RR y ) )
3	2	opabbii 3990	. . . . 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 4608	. . . . 5 |- { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ x <RR y ) } C_ ( RR X. RR )
5	3, 4	eqsstri 3124	. . . 4 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR X. RR )
6		rexpssxrxp 7803	. . . 4 |- ( RR X. RR ) C_ ( RR* X. RR* )
7	5, 6	sstri 3101	. . 3 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR* X. RR* )
8		ressxr 7802	. . . . . 6 |- RR C_ RR*
9		snsspr2 3664	. . . . . . 7 |- { -oo } C_ { +oo , -oo }
10		ssun2 3235	. . . . . . . 8 |- { +oo , -oo } C_ ( RR u. { +oo , -oo } )
11		df-xr 7797	. . . . . . . 8 |- RR* = ( RR u. { +oo , -oo } )
12	10, 11	sseqtrri 3127	. . . . . . 7 |- { +oo , -oo } C_ RR*
13	9, 12	sstri 3101	. . . . . 6 |- { -oo } C_ RR*
14	8, 13	unssi 3246	. . . . 5 |- ( RR u. { -oo } ) C_ RR*
15		snsspr1 3663	. . . . . 6 |- { +oo } C_ { +oo , -oo }
16	15, 12	sstri 3101	. . . . 5 |- { +oo } C_ RR*
17		xpss12 4641	. . . . 5 |- ( ( ( RR u. { -oo } ) C_ RR* /\ { +oo } C_ RR* ) -> ( ( RR u. { -oo } ) X. { +oo } ) C_ ( RR* X. RR* ) )
18	14, 16, 17	mp2an 422	. . . 4 |- ( ( RR u. { -oo } ) X. { +oo } ) C_ ( RR* X. RR* )
19		xpss12 4641	. . . . 5 |- ( ( { -oo } C_ RR* /\ RR C_ RR* ) -> ( { -oo } X. RR ) C_ ( RR* X. RR* ) )
20	13, 8, 19	mp2an 422	. . . 4 |- ( { -oo } X. RR ) C_ ( RR* X. RR* )
21	18, 20	unssi 3246	. . 3 |- ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) C_ ( RR* X. RR* )
22	7, 21	unssi 3246	. 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 3124	1 |- < C_ ( RR* X. RR* )

Syntax hints:   /\ wa 103   /\ w3a 962   e. wcel 1480   u. cun 3064   C_ wss 3066   {csn 3522   {cpr 3523   class class class wbr 3924   {copab 3983   X. cxp 4532   RRcr 7612   <RR cltrr 7617   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792   < clt 7793

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pr 3529  df-opab 3985  df-xp 4540  df-xr 7797  df-ltxr 7798

This theorem is referenced by:  ltrel  7819