Theorem icoiccdif 37721

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 11746	. . . . . . 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 3418	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. ( A [,] B ) )
4		elico1 11704	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
5	4	biimpa 492	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> ( x e. RR* /\ A <_ x /\ x < B ) )
6	5	simp1d 1042	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. RR* )
7		simplr 770	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> B e. RR* )
8	5	simp3d 1044	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x < B )
9		xrltne 11483	. . . . . . . . 9 |- ( ( x e. RR* /\ B e. RR* /\ x < B ) -> B =/= x )
10	6, 7, 8, 9	syl3anc 1292	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> B =/= x )
11	10	necomd 2698	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x =/= B )
12	11	neneqd 2648	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> -. x = B )
13		elsn 3973	. . . . . 6 |- ( x e. { B } <-> x = B )
14	12, 13	sylnibr 312	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> -. x e. { B } )
15	3, 14	eldifd 3401	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. ( ( A [,] B ) \ { B } ) )
16	15	ex 441	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) -> x e. ( ( A [,] B ) \ { B } ) ) )
17	16	ssrdv 3424	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) C_ ( ( A [,] B ) \ { B } ) )
18		simpll 768	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> A e. RR* )
19		simplr 770	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> B e. RR* )
20		eldifi 3544	. . . . . . 7 |- ( x e. ( ( A [,] B ) \ { B } ) -> x e. ( A [,] B ) )
21		eliccxr 37708	. . . . . . 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 473	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. RR* )
24	20	adantl 473	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. ( A [,] B ) )
25		elicc1 11705	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ x /\ x <_ B ) ) )
26	25	adantr 472	. . . . . . 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 215	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> ( x e. RR* /\ A <_ x /\ x <_ B ) )
28	27	simp2d 1043	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> A <_ x )
29		eldifsni 4089	. . . . . . . 8 |- ( x e. ( ( A [,] B ) \ { B } ) -> x =/= B )
30	29	necomd 2698	. . . . . . 7 |- ( x e. ( ( A [,] B ) \ { B } ) -> B =/= x )
31	30	adantl 473	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> B =/= x )
32	27	simp3d 1044	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x <_ B )
33		xrleltne 11467	. . . . . . 7 |- ( ( x e. RR* /\ B e. RR* /\ x <_ B ) -> ( x < B <-> B =/= x ) )
34	23, 19, 32, 33	syl3anc 1292	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> ( x < B <-> B =/= x ) )
35	31, 34	mpbird 240	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x < B )
36	18, 19, 23, 28, 35	elicod 11710	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. ( A [,) B ) )
37	36	ex 441	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( ( A [,] B ) \ { B } ) -> x e. ( A [,) B ) ) )
38	37	ssrdv 3424	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,] B ) \ { B } ) C_ ( A [,) B ) )
39	17, 38	eqssd 3435	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) = ( ( A [,] B ) \ { B } ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   \ cdif 3387   C_ wss 3390   {csn 3959   class class class wbr 4395  (class class class)co 6308   RR*cxr 9692   < clt 9693   <_ cle 9694   [,)cico 11662   [,]cicc 11663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-pre-lttri 9631  ax-pre-lttrn 9632

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-ico 11666  df-icc 11667

This theorem is referenced by: (None)