Theorem ioojoin 7585

Description: Join two open intervals to create a third.

Assertion

Ref	Expression
ioojoin	|- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> (((A(,)B) u. {B}) u. (B(,)C)) = (A(,)C))

Proof of Theorem ioojoin

Step	Hyp	Ref	Expression
1		snunioo 7584	. . . . . . 7 |- ((B e. RR /\ C e. RR /\ B < C) -> ({B} u. (B(,)C)) = (B[,)C))
2	1	3expa 1067	. . . . . 6 |- (((B e. RR /\ C e. RR) /\ B < C) -> ({B} u. (B(,)C)) = (B[,)C))
3	2	3adantl1 1032	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ B < C) -> ({B} u. (B(,)C)) = (B[,)C))
4	3	adantrl 430	. . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ({B} u. (B(,)C)) = (B[,)C))
5	4	uneq2d 2755	. . 3 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ((A(,)B) u. ({B} u. (B(,)C))) = ((A(,)B) u. (B[,)C)))
6		elioo2 7546	. . . . . . . . . . . 12 |- ((A e. RR* /\ B e. RR*) -> (x e. (A(,)B) <-> (x e. RR /\ A < x /\ x < B)))
7		rexr 6668	. . . . . . . . . . . 12 |- (A e. RR -> A e. RR*)
8		rexr 6668	. . . . . . . . . . . 12 |- (B e. RR -> B e. RR*)
9	6, 7, 8	syl2an 503	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR) -> (x e. (A(,)B) <-> (x e. RR /\ A < x /\ x < B)))
10	9	3adant3 896	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (x e. (A(,)B) <-> (x e. RR /\ A < x /\ x < B)))
11		3anass 862	. . . . . . . . . 10 |- ((x e. RR /\ A < x /\ x < B) <-> (x e. RR /\ (A < x /\ x < B)))
12	10, 11	syl6bb 595	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (x e. (A(,)B) <-> (x e. RR /\ (A < x /\ x < B))))
13		elico2 7559	. . . . . . . . . . 11 |- ((B e. RR /\ C e. RR) -> (x e. (B[,)C) <-> (x e. RR /\ B <_ x /\ x < C)))
14	13	3adant1 894	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (x e. (B[,)C) <-> (x e. RR /\ B <_ x /\ x < C)))
15		3anass 862	. . . . . . . . . 10 |- ((x e. RR /\ B <_ x /\ x < C) <-> (x e. RR /\ (B <_ x /\ x < C)))
16	14, 15	syl6bb 595	. . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (x e. (B[,)C) <-> (x e. RR /\ (B <_ x /\ x < C))))
17	12, 16	orbi12d 689	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((x e. (A(,)B) \/ x e. (B[,)C)) <-> ((x e. RR /\ (A < x /\ x < B)) \/ (x e. RR /\ (B <_ x /\ x < C)))))
18	17	adantr 425	. . . . . . 7 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ((x e. (A(,)B) \/ x e. (B[,)C)) <-> ((x e. RR /\ (A < x /\ x < B)) \/ (x e. RR /\ (B <_ x /\ x < C)))))
19		pm2.24 95	. . . . . . . . . . . . . . . . 17 |- (A < x -> (-. A < x -> B <_ x))
20	19	ad2antrl 442	. . . . . . . . . . . . . . . 16 |- (((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) /\ (A < x /\ x < C)) -> (-. A < x -> B <_ x))
21		lenlt 6679	. . . . . . . . . . . . . . . . . . . 20 |- ((B e. RR /\ x e. RR) -> (B <_ x <-> -. x < B))
22	21	biimprd 171	. . . . . . . . . . . . . . . . . . 19 |- ((B e. RR /\ x e. RR) -> (-. x < B -> B <_ x))
23	22	3ad2antl2 1039	. . . . . . . . . . . . . . . . . 18 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ x e. RR) -> (-. x < B -> B <_ x))
24	23	adantlr 429	. . . . . . . . . . . . . . . . 17 |- ((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) -> (-. x < B -> B <_ x))
25	24	adantr 425	. . . . . . . . . . . . . . . 16 |- (((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) /\ (A < x /\ x < C)) -> (-. x < B -> B <_ x))
26	20, 25	jaod 469	. . . . . . . . . . . . . . 15 |- (((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) /\ (A < x /\ x < C)) -> ((-. A < x \/ -. x < B) -> B <_ x))
27		ianor 329	. . . . . . . . . . . . . . 15 |- (-. (A < x /\ x < B) <-> (-. A < x \/ -. x < B))
28	26, 27	syl5ib 223	. . . . . . . . . . . . . 14 |- (((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) /\ (A < x /\ x < C)) -> (-. (A < x /\ x < B) -> B <_ x))
29		simpr 350	. . . . . . . . . . . . . . . 16 |- ((A < x /\ x < C) -> x < C)
30	29	a1d 15	. . . . . . . . . . . . . . 15 |- ((A < x /\ x < C) -> (-. (A < x /\ x < B) -> x < C))
31	30	adantl 424	. . . . . . . . . . . . . 14 |- (((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) /\ (A < x /\ x < C)) -> (-. (A < x /\ x < B) -> x < C))
32	28, 31	jcad 661	. . . . . . . . . . . . 13 |- (((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) /\ (A < x /\ x < C)) -> (-. (A < x /\ x < B) -> (B <_ x /\ x < C)))
33	32	orrd 250	. . . . . . . . . . . 12 |- (((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) /\ (A < x /\ x < C)) -> ((A < x /\ x < B) \/ (B <_ x /\ x < C)))
34	33	ex 402	. . . . . . . . . . 11 |- ((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) -> ((A < x /\ x < C) -> ((A < x /\ x < B) \/ (B <_ x /\ x < C))))
35		axlttrn 6673	. . . . . . . . . . . . . . . . . . . . 21 |- ((x e. RR /\ B e. RR /\ C e. RR) -> ((x < B /\ B < C) -> x < C))
36	35	exp3a 405	. . . . . . . . . . . . . . . . . . . 20 |- ((x e. RR /\ B e. RR /\ C e. RR) -> (x < B -> (B < C -> x < C)))
37	36	com23 36	. . . . . . . . . . . . . . . . . . 19 |- ((x e. RR /\ B e. RR /\ C e. RR) -> (B < C -> (x < B -> x < C)))
38	37	3exp 1066	. . . . . . . . . . . . . . . . . 18 |- (x e. RR -> (B e. RR -> (C e. RR -> (B < C -> (x < B -> x < C)))))
39	38	com4l 43	. . . . . . . . . . . . . . . . 17 |- (B e. RR -> (C e. RR -> (B < C -> (x e. RR -> (x < B -> x < C)))))
40	39	imp31 389	. . . . . . . . . . . . . . . 16 |- (((B e. RR /\ C e. RR) /\ B < C) -> (x e. RR -> (x < B -> x < C)))
41	40	3adantl1 1032	. . . . . . . . . . . . . . 15 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ B < C) -> (x e. RR -> (x < B -> x < C)))
42	41	adantrl 430	. . . . . . . . . . . . . 14 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> (x e. RR -> (x < B -> x < C)))
43	42	imp 377	. . . . . . . . . . . . 13 |- ((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) -> (x < B -> x < C))
44	43	anim2d 620	. . . . . . . . . . . 12 |- ((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) -> ((A < x /\ x < B) -> (A < x /\ x < C)))
45		ltletr 6694	. . . . . . . . . . . . . . . . . . 19 |- ((A e. RR /\ B e. RR /\ x e. RR) -> ((A < B /\ B <_ x) -> A < x))
46	45	exp5o 1087	. . . . . . . . . . . . . . . . . 18 |- (A e. RR -> (B e. RR -> (x e. RR -> (A < B -> (B <_ x -> A < x)))))
47	46	com34 40	. . . . . . . . . . . . . . . . 17 |- (A e. RR -> (B e. RR -> (A < B -> (x e. RR -> (B <_ x -> A < x)))))
48	47	imp31 389	. . . . . . . . . . . . . . . 16 |- (((A e. RR /\ B e. RR) /\ A < B) -> (x e. RR -> (B <_ x -> A < x)))
49	48	3adantl3 1034	. . . . . . . . . . . . . . 15 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ A < B) -> (x e. RR -> (B <_ x -> A < x)))
50	49	adantrr 431	. . . . . . . . . . . . . 14 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> (x e. RR -> (B <_ x -> A < x)))
51	50	imp 377	. . . . . . . . . . . . 13 |- ((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) -> (B <_ x -> A < x))
52	51	anim1d 619	. . . . . . . . . . . 12 |- ((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) -> ((B <_ x /\ x < C) -> (A < x /\ x < C)))
53	44, 52	jaod 469	. . . . . . . . . . 11 |- ((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) -> (((A < x /\ x < B) \/ (B <_ x /\ x < C)) -> (A < x /\ x < C)))
54	34, 53	impbid 574	. . . . . . . . . 10 |- ((((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) /\ x e. RR) -> ((A < x /\ x < C) <-> ((A < x /\ x < B) \/ (B <_ x /\ x < C))))
55	54	pm5.32da 711	. . . . . . . . 9 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ((x e. RR /\ (A < x /\ x < C)) <-> (x e. RR /\ ((A < x /\ x < B) \/ (B <_ x /\ x < C)))))
56		3anass 862	. . . . . . . . 9 |- ((x e. RR /\ A < x /\ x < C) <-> (x e. RR /\ (A < x /\ x < C)))
57	55, 56	syl5bb 591	. . . . . . . 8 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ((x e. RR /\ A < x /\ x < C) <-> (x e. RR /\ ((A < x /\ x < B) \/ (B <_ x /\ x < C)))))
58		andi 665	. . . . . . . 8 |- ((x e. RR /\ ((A < x /\ x < B) \/ (B <_ x /\ x < C))) <-> ((x e. RR /\ (A < x /\ x < B)) \/ (x e. RR /\ (B <_ x /\ x < C))))
59	57, 58	syl6bb 595	. . . . . . 7 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ((x e. RR /\ A < x /\ x < C) <-> ((x e. RR /\ (A < x /\ x < B)) \/ (x e. RR /\ (B <_ x /\ x < C)))))
60	18, 59	bitr4d 590	. . . . . 6 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ((x e. (A(,)B) \/ x e. (B[,)C)) <-> (x e. RR /\ A < x /\ x < C)))
61		elun 2741	. . . . . 6 |- (x e. ((A(,)B) u. (B[,)C)) <-> (x e. (A(,)B) \/ x e. (B[,)C)))
62	60, 61	syl5bb 591	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> (x e. ((A(,)B) u. (B[,)C)) <-> (x e. RR /\ A < x /\ x < C)))
63		elioo2 7546	. . . . . . . 8 |- ((A e. RR* /\ C e. RR*) -> (x e. (A(,)C) <-> (x e. RR /\ A < x /\ x < C)))
64		rexr 6668	. . . . . . . 8 |- (C e. RR -> C e. RR*)
65	63, 7, 64	syl2an 503	. . . . . . 7 |- ((A e. RR /\ C e. RR) -> (x e. (A(,)C) <-> (x e. RR /\ A < x /\ x < C)))
66	65	3adant2 895	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (x e. (A(,)C) <-> (x e. RR /\ A < x /\ x < C)))
67	66	adantr 425	. . . . 5 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> (x e. (A(,)C) <-> (x e. RR /\ A < x /\ x < C)))
68	62, 67	bitr4d 590	. . . 4 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> (x e. ((A(,)B) u. (B[,)C)) <-> x e. (A(,)C)))
69	68	eqrdv 1882	. . 3 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ((A(,)B) u. (B[,)C)) = (A(,)C))
70	5, 69	eqtrd 1925	. 2 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> ((A(,)B) u. ({B} u. (B(,)C))) = (A(,)C))
71		unass 2759	. 2 |- (((A(,)B) u. {B}) u. (B(,)C)) = ((A(,)B) u. ({B} u. (B(,)C)))
72	70, 71	syl5eq 1940	1 |- (((A e. RR /\ B e. RR /\ C e. RR) /\ (A < B /\ B < C)) -> (((A(,)B) u. {B}) u. (B(,)C)) = (A(,)C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   u. cun 2591  {csn 3044   class class class wbr 3338  (class class class)co 4884  RRcr 6385   <_ cle 6448  RR*cxr 6652   < clt 6653  (,)cioo 7524  [,)cico 7526

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-ioo 7528  df-ico 7530

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