Theorem snunioolem 7583

Description: Lemma for snunioo 7584.

Hypotheses

Ref	Expression
snunioolem.1	|- A e. RR
snunioolem.2	|- B e. RR

Assertion

Ref	Expression
snunioolem	|- (A < B -> ({A} u. (A(,)B)) = (A[,)B))

Proof of Theorem snunioolem

Step	Hyp	Ref	Expression
1		snunioolem.1	. . . . . . . . . . 11 |- A e. RR
2		leloe 6688	. . . . . . . . . . 11 |- ((A e. RR /\ x e. RR) -> (A <_ x <-> (A < x \/ A = x)))
3	1, 2	mpan 759	. . . . . . . . . 10 |- (x e. RR -> (A <_ x <-> (A < x \/ A = x)))
4	3	anbi1d 679	. . . . . . . . 9 |- (x e. RR -> ((A <_ x /\ x < B) <-> ((A < x \/ A = x) /\ x < B)))
5		breq1 3341	. . . . . . . . . . . . 13 |- (A = x -> (A < B <-> x < B))
6	5	biimpcd 172	. . . . . . . . . . . 12 |- (A < B -> (A = x -> x < B))
7		pm4.71 697	. . . . . . . . . . . 12 |- ((A = x -> x < B) <-> (A = x <-> (A = x /\ x < B)))
8	6, 7	sylib 215	. . . . . . . . . . 11 |- (A < B -> (A = x <-> (A = x /\ x < B)))
9	8	orbi2d 676	. . . . . . . . . 10 |- (A < B -> (((A < x /\ x < B) \/ A = x) <-> ((A < x /\ x < B) \/ (A = x /\ x < B))))
10		andir 666	. . . . . . . . . 10 |- (((A < x \/ A = x) /\ x < B) <-> ((A < x /\ x < B) \/ (A = x /\ x < B)))
11	9, 10	syl6rbbr 598	. . . . . . . . 9 |- (A < B -> (((A < x \/ A = x) /\ x < B) <-> ((A < x /\ x < B) \/ A = x)))
12	4, 11	sylan9bb 599	. . . . . . . 8 |- ((x e. RR /\ A < B) -> ((A <_ x /\ x < B) <-> ((A < x /\ x < B) \/ A = x)))
13	12	expcom 403	. . . . . . 7 |- (A < B -> (x e. RR -> ((A <_ x /\ x < B) <-> ((A < x /\ x < B) \/ A = x))))
14	13	pm5.32d 709	. . . . . 6 |- (A < B -> ((x e. RR /\ (A <_ x /\ x < B)) <-> (x e. RR /\ ((A < x /\ x < B) \/ A = x))))
15		andi 665	. . . . . . 7 |- ((x e. RR /\ ((A < x /\ x < B) \/ A = x)) <-> ((x e. RR /\ (A < x /\ x < B)) \/ (x e. RR /\ A = x)))
16		snunioolem.2	. . . . . . . . . 10 |- B e. RR
17		elioo2 7546	. . . . . . . . . . 11 |- ((A e. RR* /\ B e. RR*) -> (x e. (A(,)B) <-> (x e. RR /\ A < x /\ x < B)))
18		rexr 6668	. . . . . . . . . . 11 |- (A e. RR -> A e. RR*)
19		rexr 6668	. . . . . . . . . . 11 |- (B e. RR -> B e. RR*)
20	17, 18, 19	syl2an 503	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR) -> (x e. (A(,)B) <-> (x e. RR /\ A < x /\ x < B)))
21	1, 16, 20	mp2an 761	. . . . . . . . 9 |- (x e. (A(,)B) <-> (x e. RR /\ A < x /\ x < B))
22		3anass 862	. . . . . . . . 9 |- ((x e. RR /\ A < x /\ x < B) <-> (x e. RR /\ (A < x /\ x < B)))
23	21, 22	bitri 190	. . . . . . . 8 |- (x e. (A(,)B) <-> (x e. RR /\ (A < x /\ x < B)))
24		eleq1 1957	. . . . . . . . . 10 |- (A = x -> (A e. RR <-> x e. RR))
25	1, 24	mpbii 210	. . . . . . . . 9 |- (A = x -> x e. RR)
26	25	pm4.71ri 700	. . . . . . . 8 |- (A = x <-> (x e. RR /\ A = x))
27	23, 26	orbi12i 277	. . . . . . 7 |- ((x e. (A(,)B) \/ A = x) <-> ((x e. RR /\ (A < x /\ x < B)) \/ (x e. RR /\ A = x)))
28	15, 27	bitr4i 193	. . . . . 6 |- ((x e. RR /\ ((A < x /\ x < B) \/ A = x)) <-> (x e. (A(,)B) \/ A = x))
29	14, 28	syl6bb 595	. . . . 5 |- (A < B -> ((x e. RR /\ (A <_ x /\ x < B)) <-> (x e. (A(,)B) \/ A = x)))
30		elico2 7559	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (x e. (A[,)B) <-> (x e. RR /\ A <_ x /\ x < B)))
31	1, 16, 30	mp2an 761	. . . . . 6 |- (x e. (A[,)B) <-> (x e. RR /\ A <_ x /\ x < B))
32		3anass 862	. . . . . 6 |- ((x e. RR /\ A <_ x /\ x < B) <-> (x e. RR /\ (A <_ x /\ x < B)))
33	31, 32	bitri 190	. . . . 5 |- (x e. (A[,)B) <-> (x e. RR /\ (A <_ x /\ x < B)))
34	29, 33	syl5bb 591	. . . 4 |- (A < B -> (x e. (A[,)B) <-> (x e. (A(,)B) \/ A = x)))
35		orcom 266	. . . 4 |- ((x e. (A(,)B) \/ A = x) <-> (A = x \/ x e. (A(,)B)))
36	34, 35	syl6bb 595	. . 3 |- (A < B -> (x e. (A[,)B) <-> (A = x \/ x e. (A(,)B))))
37		elun 2741	. . . 4 |- (x e. ({A} u. (A(,)B)) <-> (x e. {A} \/ x e. (A(,)B)))
38		elsn 3058	. . . . . 6 |- (x e. {A} <-> x = A)
39		eqcom 1886	. . . . . 6 |- (x = A <-> A = x)
40	38, 39	bitri 190	. . . . 5 |- (x e. {A} <-> A = x)
41	40	orbi1i 276	. . . 4 |- ((x e. {A} \/ x e. (A(,)B)) <-> (A = x \/ x e. (A(,)B)))
42	37, 41	bitri 190	. . 3 |- (x e. ({A} u. (A(,)B)) <-> (A = x \/ x e. (A(,)B)))
43	36, 42	syl6rbbr 598	. 2 |- (A < B -> (x e. ({A} u. (A(,)B)) <-> x e. (A[,)B)))
44	43	eqrdv 1882	1 |- (A < B -> ({A} u. (A(,)B)) = (A[,)B))

Syntax hints:   -> 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 is referenced by:  snunioo 7584

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-ltp 6242  df-enr 6318  df-nr 6319  df-ltr 6322  df-0r 6323  df-c 6392  df-r 6396  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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