Theorem ltrelxr 8017

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 7996	. 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 980	. . . . . 6 |- ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( ( x e. RR /\ y e. RR ) /\ x <RR y ) )
3	2	opabbii 4070	. . . . 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 4700	. . . . 5 |- { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ x <RR y ) } C_ ( RR X. RR )
5	3, 4	eqsstri 3187	. . . 4 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR X. RR )
6		rexpssxrxp 8001	. . . 4 |- ( RR X. RR ) C_ ( RR* X. RR* )
7	5, 6	sstri 3164	. . 3 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR* X. RR* )
8		ressxr 8000	. . . . . 6 |- RR C_ RR*
9		snsspr2 3741	. . . . . . 7 |- { -oo } C_ { +oo , -oo }
10		ssun2 3299	. . . . . . . 8 |- { +oo , -oo } C_ ( RR u. { +oo , -oo } )
11		df-xr 7995	. . . . . . . 8 |- RR* = ( RR u. { +oo , -oo } )
12	10, 11	sseqtrri 3190	. . . . . . 7 |- { +oo , -oo } C_ RR*
13	9, 12	sstri 3164	. . . . . 6 |- { -oo } C_ RR*
14	8, 13	unssi 3310	. . . . 5 |- ( RR u. { -oo } ) C_ RR*
15		snsspr1 3740	. . . . . 6 |- { +oo } C_ { +oo , -oo }
16	15, 12	sstri 3164	. . . . 5 |- { +oo } C_ RR*
17		xpss12 4733	. . . . 5 |- ( ( ( RR u. { -oo } ) C_ RR* /\ { +oo } C_ RR* ) -> ( ( RR u. { -oo } ) X. { +oo } ) C_ ( RR* X. RR* ) )
18	14, 16, 17	mp2an 426	. . . 4 |- ( ( RR u. { -oo } ) X. { +oo } ) C_ ( RR* X. RR* )
19		xpss12 4733	. . . . 5 |- ( ( { -oo } C_ RR* /\ RR C_ RR* ) -> ( { -oo } X. RR ) C_ ( RR* X. RR* ) )
20	13, 8, 19	mp2an 426	. . . 4 |- ( { -oo } X. RR ) C_ ( RR* X. RR* )
21	18, 20	unssi 3310	. . 3 |- ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) C_ ( RR* X. RR* )
22	7, 21	unssi 3310	. 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 3187	1 |- < C_ ( RR* X. RR* )

Syntax hints:   /\ wa 104   /\ w3a 978   e. wcel 2148   u. cun 3127   C_ wss 3129   {csn 3592   {cpr 3593   class class class wbr 4003   {copab 4063   X. cxp 4624   RRcr 7809   <RR cltrr 7814   +oocpnf 7988   -oocmnf 7989   RR*cxr 7990   < clt 7991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pr 3599  df-opab 4065  df-xp 4632  df-xr 7995  df-ltxr 7996

This theorem is referenced by:  ltrel  8018