Theorem elicore 10223

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 9851	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
2	1	elixx3g 9858	. . . . . 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 1006	. . 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 1005	. . . 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 7972	. . . 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 9746	. . . . 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 9768	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C < +oo )
21		xrre3 9779	. 2 |- ( ( ( C e. RR* /\ A e. RR ) /\ ( A <_ C /\ C < +oo ) ) -> C e. RR )
22	6, 7, 10, 20, 21	syl22anc 1234	1 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   +oocpnf 7951   RR*cxr 7953   < clt 7954   <_ cle 7955   [,)cico 9847

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-ico 9851

This theorem is referenced by: (None)