Theorem elicore 10481

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 10086	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
2	1	elixx3g 10093	. . . . . 6 |- ( C e. ( A [,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ C /\ C < B ) ) )
3	2	biimpi 120	. . . . 5 |- ( C e. ( A [,) B ) -> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A <_ C /\ C < B ) ) )
4	3	simpld 112	. . . 4 |- ( C e. ( A [,) B ) -> ( A e. RR* /\ B e. RR* /\ C e. RR* ) )
5	4	simp3d 1035	. . 3 |- ( C e. ( A [,) B ) -> C e. RR* )
6	5	adantl 277	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR* )
7		simpl 109	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> A e. RR )
8	3	simprd 114	. . . 4 |- ( C e. ( A [,) B ) -> ( A <_ C /\ C < B ) )
9	8	simpld 112	. . 3 |- ( C e. ( A [,) B ) -> A <_ C )
10	9	adantl 277	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> A <_ C )
11	4	simp2d 1034	. . . 4 |- ( C e. ( A [,) B ) -> B e. RR* )
12	11	adantl 277	. . 3 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> B e. RR* )
13		pnfxr 8195	. . . 4 |- +oo e. RR*
14	13	a1i 9	. . 3 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> +oo e. RR* )
15	8	simprd 114	. . . 4 |- ( C e. ( A [,) B ) -> C < B )
16	15	adantl 277	. . 3 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C < B )
17		pnfge 9981	. . . . 5 |- ( B e. RR* -> B <_ +oo )
18	11, 17	syl 14	. . . 4 |- ( C e. ( A [,) B ) -> B <_ +oo )
19	18	adantl 277	. . 3 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> B <_ +oo )
20	6, 12, 14, 16, 19	xrltletrd 10003	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C < +oo )
21		xrre3 10014	. 2 |- ( ( ( C e. RR* /\ A e. RR ) /\ ( A <_ C /\ C < +oo ) ) -> C e. RR )
22	6, 7, 10, 20, 21	syl22anc 1272	1 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   RRcr 7994   +oocpnf 8174   RR*cxr 8176   < clt 8177   <_ cle 8178   [,)cico 10082

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-ico 10086

This theorem is referenced by: (None)