Theorem icoiccdif 39750

Description: Left closed, right open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
icoiccdif	|- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) = ( ( A [,] B ) \ { B } ) )

Proof of Theorem icoiccdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icossicc 12260	. . . . . . 7 |- ( A [,) B ) C_ ( A [,] B )
2	1	a1i 11	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) C_ ( A [,] B ) )
3	2	sselda 3603	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. ( A [,] B ) )
4		elico1 12218	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
5	4	biimpa 501	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> ( x e. RR* /\ A <_ x /\ x < B ) )
6	5	simp1d 1073	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. RR* )
7		simplr 792	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> B e. RR* )
8	5	simp3d 1075	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x < B )
9		xrltne 11994	. . . . . . . . 9 |- ( ( x e. RR* /\ B e. RR* /\ x < B ) -> B =/= x )
10	6, 7, 8, 9	syl3anc 1326	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> B =/= x )
11	10	necomd 2849	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x =/= B )
12	11	neneqd 2799	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> -. x = B )
13		velsn 4193	. . . . . 6 |- ( x e. { B } <-> x = B )
14	12, 13	sylnibr 319	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> -. x e. { B } )
15	3, 14	eldifd 3585	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. ( ( A [,] B ) \ { B } ) )
16	15	ex 450	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) -> x e. ( ( A [,] B ) \ { B } ) ) )
17	16	ssrdv 3609	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) C_ ( ( A [,] B ) \ { B } ) )
18		simpll 790	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> A e. RR* )
19		simplr 792	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> B e. RR* )
20		eldifi 3732	. . . . . . 7 |- ( x e. ( ( A [,] B ) \ { B } ) -> x e. ( A [,] B ) )
21		eliccxr 39737	. . . . . . 7 |- ( x e. ( A [,] B ) -> x e. RR* )
22	20, 21	syl 17	. . . . . 6 |- ( x e. ( ( A [,] B ) \ { B } ) -> x e. RR* )
23	22	adantl 482	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. RR* )
24	20	adantl 482	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. ( A [,] B ) )
25		elicc1 12219	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ x /\ x <_ B ) ) )
26	25	adantr 481	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ x /\ x <_ B ) ) )
27	24, 26	mpbid 222	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> ( x e. RR* /\ A <_ x /\ x <_ B ) )
28	27	simp2d 1074	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> A <_ x )
29		eldifsni 4320	. . . . . . . 8 |- ( x e. ( ( A [,] B ) \ { B } ) -> x =/= B )
30	29	necomd 2849	. . . . . . 7 |- ( x e. ( ( A [,] B ) \ { B } ) -> B =/= x )
31	30	adantl 482	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> B =/= x )
32	27	simp3d 1075	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x <_ B )
33		xrleltne 11978	. . . . . . 7 |- ( ( x e. RR* /\ B e. RR* /\ x <_ B ) -> ( x < B <-> B =/= x ) )
34	23, 19, 32, 33	syl3anc 1326	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> ( x < B <-> B =/= x ) )
35	31, 34	mpbird 247	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x < B )
36	18, 19, 23, 28, 35	elicod 12224	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. ( A [,) B ) )
37	36	ex 450	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( ( A [,] B ) \ { B } ) -> x e. ( A [,) B ) ) )
38	37	ssrdv 3609	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,] B ) \ { B } ) C_ ( A [,) B ) )
39	17, 38	eqssd 3620	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) = ( ( A [,] B ) \ { B } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177   [,]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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181  df-icc 12182

This theorem is referenced by: (None)