Theorem elicore 10159

Description: A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
elicore	|- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR )

Proof of Theorem elicore

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

Step	Hyp	Ref	Expression
1		df-ico 9791	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
2	1	elixx3g 9798	. . . . . 6 |- ( C e. ( A [,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ C /\ C < B ) ) )
3	2	biimpi 119	. . . . 5 |- ( C e. ( A [,) B ) -> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ C /\ C < B ) ) )
4	3	simpld 111	. . . 4 |- ( C e. ( A [,) B ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
5	4	simp3d 996	. . 3 |- ( C e. ( A [,) B ) -> C e. RR* )
6	5	adantl 275	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR* )
7		simpl 108	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> A e. RR )
8	3	simprd 113	. . . 4 |- ( C e. ( A [,) B ) -> ( A <_ C /\ C < B ) )
9	8	simpld 111	. . 3 |- ( C e. ( A [,) B ) -> A <_ C )
10	9	adantl 275	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> A <_ C )
11	4	simp2d 995	. . . 4 |- ( C e. ( A [,) B ) -> B e. RR* )
12	11	adantl 275	. . 3 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> B e. RR* )
13		pnfxr 7924	. . . 4 |- +oo e. RR*
14	13	a1i 9	. . 3 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> +oo e. RR* )
15	8	simprd 113	. . . 4 |- ( C e. ( A [,) B ) -> C < B )
16	15	adantl 275	. . 3 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C < B )
17		pnfge 9689	. . . . 5 |- ( B e. RR* -> B <_ +oo )
18	11, 17	syl 14	. . . 4 |- ( C e. ( A [,) B ) -> B <_ +oo )
19	18	adantl 275	. . 3 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> B <_ +oo )
20	6, 12, 14, 16, 19	xrltletrd 9708	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C < +oo )
21		xrre3 9719	. 2 |- ( ( ( C e. RR* /\ A e. RR ) /\ ( A <_ C /\ C < +oo ) ) -> C e. RR )
22	6, 7, 10, 20, 21	syl22anc 1221	1 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   e. wcel 2128   class class class wbr 3965  (class class class)co 5821   RRcr 7725   +oocpnf 7903   RR*cxr 7905   < clt 7906   <_ cle 7907   [,)cico 9787

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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7817  ax-resscn 7818  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-ico 9791

This theorem is referenced by: (None)