Theorem icomnfordt 21020

Description: An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
icomnfordt	|- ( -oo [,) A ) e. (ordTop ` <_ )

Proof of Theorem icomnfordt

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . 9 |- ran ( x e. RR* |-> ( x (,] +oo ) ) = ran ( x e. RR* |-> ( x (,] +oo ) )
2		eqid 2622	. . . . . . . . 9 |- ran ( x e. RR* |-> ( -oo [,) x ) ) = ran ( x e. RR* |-> ( -oo [,) x ) )
3		eqid 2622	. . . . . . . . 9 |- ran (,) = ran (,)
4	1, 2, 3	leordtval 21017	. . . . . . . 8 |- (ordTop ` <_ ) = ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) )
5		letop 21010	. . . . . . . 8 |- (ordTop ` <_ ) e. Top
6	4, 5	eqeltrri 2698	. . . . . . 7 |- ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) e. Top
7		tgclb 20774	. . . . . . 7 |- ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) e. TopBases <-> ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) e. Top )
8	6, 7	mpbir 221	. . . . . 6 |- ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) e. TopBases
9		bastg 20770	. . . . . 6 |- ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) e. TopBases -> ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) C_ ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) )
10	8, 9	ax-mp 5	. . . . 5 |- ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) C_ ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) )
11	10, 4	sseqtr4i 3638	. . . 4 |- ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) C_ (ordTop ` <_ )
12		ssun1 3776	. . . . 5 |- ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) C_ ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) )
13		ssun2 3777	. . . . . 6 |- ran ( x e. RR* |-> ( -oo [,) x ) ) C_ ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) )
14		eqid 2622	. . . . . . . 8 |- ( -oo [,) A ) = ( -oo [,) A )
15		oveq2 6658	. . . . . . . . . 10 |- ( x = A -> ( -oo [,) x ) = ( -oo [,) A ) )
16	15	eqeq2d 2632	. . . . . . . . 9 |- ( x = A -> ( ( -oo [,) A ) = ( -oo [,) x ) <-> ( -oo [,) A ) = ( -oo [,) A ) ) )
17	16	rspcev 3309	. . . . . . . 8 |- ( ( A e. RR* /\ ( -oo [,) A ) = ( -oo [,) A ) ) -> E. x e. RR* ( -oo [,) A ) = ( -oo [,) x ) )
18	14, 17	mpan2 707	. . . . . . 7 |- ( A e. RR* -> E. x e. RR* ( -oo [,) A ) = ( -oo [,) x ) )
19		eqid 2622	. . . . . . . 8 |- ( x e. RR* |-> ( -oo [,) x ) ) = ( x e. RR* |-> ( -oo [,) x ) )
20		ovex 6678	. . . . . . . 8 |- ( -oo [,) x ) e. _V
21	19, 20	elrnmpti 5376	. . . . . . 7 |- ( ( -oo [,) A ) e. ran ( x e. RR* |-> ( -oo [,) x ) ) <-> E. x e. RR* ( -oo [,) A ) = ( -oo [,) x ) )
22	18, 21	sylibr 224	. . . . . 6 |- ( A e. RR* -> ( -oo [,) A ) e. ran ( x e. RR* |-> ( -oo [,) x ) ) )
23	13, 22	sseldi 3601	. . . . 5 |- ( A e. RR* -> ( -oo [,) A ) e. ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) )
24	12, 23	sseldi 3601	. . . 4 |- ( A e. RR* -> ( -oo [,) A ) e. ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) )
25	11, 24	sseldi 3601	. . 3 |- ( A e. RR* -> ( -oo [,) A ) e. (ordTop ` <_ ) )
26	25	adantl 482	. 2 |- ( ( -oo e. RR* /\ A e. RR* ) -> ( -oo [,) A ) e. (ordTop ` <_ ) )
27		df-ico 12181	. . . . . 6 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
28	27	ixxf 12185	. . . . 5 |- [,) : ( RR* X. RR* ) --> ~P RR*
29	28	fdmi 6052	. . . 4 |- dom [,) = ( RR* X. RR* )
30	29	ndmov 6818	. . 3 |- ( -. ( -oo e. RR* /\ A e. RR* ) -> ( -oo [,) A ) = (/) )
31		0opn 20709	. . . 4 |- ( (ordTop ` <_ ) e. Top -> (/) e. (ordTop ` <_ ) )
32	5, 31	ax-mp 5	. . 3 |- (/) e. (ordTop ` <_ )
33	30, 32	syl6eqel 2709	. 2 |- ( -. ( -oo e. RR* /\ A e. RR* ) -> ( -oo [,) A ) e. (ordTop ` <_ ) )
34	26, 33	pm2.61i 176	1 |- ( -oo [,) A ) e. (ordTop ` <_ )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   u. cun 3572   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |-> cmpt 4729   X. cxp 5112   ran crn 5115   ` cfv 5888  (class class class)co 6650   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   (,]cioc 12176   [,)cico 12177   topGenctg 16098  ordTopcordt 16159   Topctop 20698   TopBasesctb 20749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-topgen 16104  df-ordt 16161  df-ps 17200  df-tsr 17201  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by:  xlimmnfvlem1  40058