Theorem ltrelxr 8133

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 8112	. 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 983	. . . . . 6 |- ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( ( x e. RR /\ y e. RR ) /\ x <RR y ) )
3	2	opabbii 4111	. . . . 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 4749	. . . . 5 |- { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ x <RR y ) } C_ ( RR X. RR )
5	3, 4	eqsstri 3225	. . . 4 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR X. RR )
6		rexpssxrxp 8117	. . . 4 |- ( RR X. RR ) C_ ( RR* X. RR* )
7	5, 6	sstri 3202	. . 3 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } C_ ( RR* X. RR* )
8		ressxr 8116	. . . . . 6 |- RR C_ RR*
9		snsspr2 3782	. . . . . . 7 |- { -oo } C_ { +oo , -oo }
10		ssun2 3337	. . . . . . . 8 |- { +oo , -oo } C_ ( RR u. { +oo , -oo } )
11		df-xr 8111	. . . . . . . 8 |- RR* = ( RR u. { +oo , -oo } )
12	10, 11	sseqtrri 3228	. . . . . . 7 |- { +oo , -oo } C_ RR*
13	9, 12	sstri 3202	. . . . . 6 |- { -oo } C_ RR*
14	8, 13	unssi 3348	. . . . 5 |- ( RR u. { -oo } ) C_ RR*
15		snsspr1 3781	. . . . . 6 |- { +oo } C_ { +oo , -oo }
16	15, 12	sstri 3202	. . . . 5 |- { +oo } C_ RR*
17		xpss12 4782	. . . . 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 4782	. . . . 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 3348	. . 3 |- ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) C_ ( RR* X. RR* )
22	7, 21	unssi 3348	. 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 3225	1 |- < C_ ( RR* X. RR* )

Syntax hints:   /\ wa 104   /\ w3a 981   e. wcel 2176   u. cun 3164   C_ wss 3166   {csn 3633   {cpr 3634   class class class wbr 4044   {copab 4104   X. cxp 4673   RRcr 7924   <RR cltrr 7929   +oocpnf 8104   -oocmnf 8105   RR*cxr 8106   < clt 8107

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pr 3640  df-opab 4106  df-xp 4681  df-xr 8111  df-ltxr 8112

This theorem is referenced by:  ltrel  8134