Theorem icoiccdif 37619

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 11718	. . . . . . 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 3431	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. ( A [,] B ) )
4		elico1 11676	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR* /\ A <_ x /\ x < B ) ) )
5	4	biimpa 487	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> ( x e. RR* /\ A <_ x /\ x < B ) )
6	5	simp1d 1019	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. RR* )
7		simplr 761	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> B e. RR* )
8	5	simp3d 1021	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x < B )
9		xrltne 11457	. . . . . . . . 9 |- ( ( x e. RR* /\ B e. RR* /\ x < B ) -> B =/= x )
10	6, 7, 8, 9	syl3anc 1267	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> B =/= x )
11	10	necomd 2678	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x =/= B )
12	11	neneqd 2628	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> -. x = B )
13		elsn 3981	. . . . . 6 |- ( x e. { B } <-> x = B )
14	12, 13	sylnibr 307	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> -. x e. { B } )
15	3, 14	eldifd 3414	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( A [,) B ) ) -> x e. ( ( A [,] B ) \ { B } ) )
16	15	ex 436	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,) B ) -> x e. ( ( A [,] B ) \ { B } ) ) )
17	16	ssrdv 3437	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) C_ ( ( A [,] B ) \ { B } ) )
18		simpll 759	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> A e. RR* )
19		simplr 761	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> B e. RR* )
20		eldifi 3554	. . . . . . 7 |- ( x e. ( ( A [,] B ) \ { B } ) -> x e. ( A [,] B ) )
21		eliccxr 37606	. . . . . . 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 468	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. RR* )
24	20	adantl 468	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. ( A [,] B ) )
25		elicc1 11677	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ x /\ x <_ B ) ) )
26	25	adantr 467	. . . . . . 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 214	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> ( x e. RR* /\ A <_ x /\ x <_ B ) )
28	27	simp2d 1020	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> A <_ x )
29		eldifsni 4097	. . . . . . . 8 |- ( x e. ( ( A [,] B ) \ { B } ) -> x =/= B )
30	29	necomd 2678	. . . . . . 7 |- ( x e. ( ( A [,] B ) \ { B } ) -> B =/= x )
31	30	adantl 468	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> B =/= x )
32	27	simp3d 1021	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x <_ B )
33		xrleltne 11441	. . . . . . 7 |- ( ( x e. RR* /\ B e. RR* /\ x <_ B ) -> ( x < B <-> B =/= x ) )
34	23, 19, 32, 33	syl3anc 1267	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> ( x < B <-> B =/= x ) )
35	31, 34	mpbird 236	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x < B )
36	18, 19, 23, 28, 35	elicod 11682	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. ( ( A [,] B ) \ { B } ) ) -> x e. ( A [,) B ) )
37	36	ex 436	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( ( A [,] B ) \ { B } ) -> x e. ( A [,) B ) ) )
38	37	ssrdv 3437	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,] B ) \ { B } ) C_ ( A [,) B ) )
39	17, 38	eqssd 3448	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) = ( ( A [,] B ) \ { B } ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   =/= wne 2621   \ cdif 3400   C_ wss 3403   {csn 3967   class class class wbr 4401  (class class class)co 6288   RR*cxr 9671   < clt 9672   <_ cle 9673   [,)cico 11634   [,]cicc 11635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-pre-lttri 9610  ax-pre-lttrn 9611

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-ico 11638  df-icc 11639

This theorem is referenced by: (None)