Theorem elicore 10409

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 10016	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
2	1	elixx3g 10023	. . . . . 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 1014	. . 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 1013	. . . 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 8125	. . . 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 9911	. . . . 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 9933	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C < +oo )
21		xrre3 9944	. 2 |- ( ( ( C e. RR* /\ A e. RR ) /\ ( A <_ C /\ C < +oo ) ) -> C e. RR )
22	6, 7, 10, 20, 21	syl22anc 1251	1 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C e. RR )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   RRcr 7924   +oocpnf 8104   RR*cxr 8106   < clt 8107   <_ cle 8108   [,)cico 10012

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 615  ax-in2 616  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-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-ico 10016

This theorem is referenced by: (None)