Theorem ltnelicc 39719

Description: A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
ltnelicc.a	|- ( ph -> A e. RR )
ltnelicc.b	|- ( ph -> B e. RR* )
ltnelicc.c	|- ( ph -> C e. RR* )
ltnelicc.clta	|- ( ph -> C < A )

Assertion

Ref	Expression
ltnelicc	|- ( ph -> -. C e. ( A [,] B ) )

Proof of Theorem ltnelicc

Step	Hyp	Ref	Expression
1		ltnelicc.clta	. . . 4 |- ( ph -> C < A )
2		ltnelicc.c	. . . . 5 |- ( ph -> C e. RR* )
3		ltnelicc.a	. . . . . 6 |- ( ph -> A e. RR )
4	3	rexrd 10089	. . . . 5 |- ( ph -> A e. RR* )
5		xrltnle 10105	. . . . 5 |- ( ( C e. RR* /\ A e. RR* ) -> ( C < A <-> -. A <_ C ) )
6	2, 4, 5	syl2anc 693	. . . 4 |- ( ph -> ( C < A <-> -. A <_ C ) )
7	1, 6	mpbid 222	. . 3 |- ( ph -> -. A <_ C )
8	7	intnanrd 963	. 2 |- ( ph -> -. ( A <_ C /\ C <_ B ) )
9		ltnelicc.b	. . 3 |- ( ph -> B e. RR* )
10		elicc4 12240	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) )
11	4, 9, 2, 10	syl3anc 1326	. 2 |- ( ph -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) )
12	8, 11	mtbird 315	1 |- ( ph -> -. C e. ( A [,] B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   [,]cicc 12178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xr 10078  df-le 10080  df-icc 12182

This theorem is referenced by:  fourierdlem104  40427