Theorem elicore 10446

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 10051	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
2	1	elixx3g 10058	. . . . . 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 8160	. . . 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 9946	. . . . 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 9968	. 2 |- ( ( A e. RR /\ C e. ( A [,) B ) ) -> C < +oo )
21		xrre3 9979	. 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 2178   class class class wbr 4059  (class class class)co 5967   RRcr 7959   +oocpnf 8139   RR*cxr 8141   < clt 8142   <_ cle 8143   [,)cico 10047

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-ico 10051

This theorem is referenced by: (None)