Theorem bsi 14845

Description: Membership to the set of open intervals implied the existence of two bounds in the set of the extended reals.

Assertion

Ref	Expression
bsi	|- (A e. ran (,) -> E.x e. RR* E.y e. RR* A = (x(,)y))

Distinct variable group:   x,A,y

Proof of Theorem bsi

Step	Hyp	Ref	Expression
1		eqeq1 1890	. . . . . . . . 9 |- (z = A -> (z = {w e. RR* | (x < w /\ w < y)} <-> A = {w e. RR* | (x < w /\ w < y)}))
2	1	anbi2d 678	. . . . . . . 8 |- (z = A -> (((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)}) <-> ((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)})))
3	2	2exbidv 1659	. . . . . . 7 |- (z = A -> (E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)}) <-> E.xE.y((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)})))
4	3	ceqsexgv 2393	. . . . . 6 |- (A e. ran (,) -> (E.z(z = A /\ E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})) <-> E.xE.y((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)})))
5	4	biimpd 170	. . . . 5 |- (A e. ran (,) -> (E.z(z = A /\ E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})) -> E.xE.y((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)})))
6		clelab 2013	. . . . 5 |- (A e. {z | E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})} <-> E.z(z = A /\ E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})))
7	5, 6	syl5ib 223	. . . 4 |- (A e. ran (,) -> (A e. {z | E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})} -> E.xE.y((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)})))
8		df-ioo 7528	. . . . . . 7 |- (,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
9	8	rneqi 4187	. . . . . 6 |- ran (,) = ran {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
10	9	eleq2i 1961	. . . . 5 |- (A e. ran (,) <-> A e. ran {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})})
11		rnoprab 4933	. . . . . 6 |- ran {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})} = {z | E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
12	11	eleq2i 1961	. . . . 5 |- (A e. ran {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})} <-> A e. {z | E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})})
13	10, 12	bitri 190	. . . 4 |- (A e. ran (,) <-> A e. {z | E.xE.y((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})})
14	7, 13	syl5ib 223	. . 3 |- (A e. ran (,) -> (A e. ran (,) -> E.xE.y((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)})))
15		iooval 7533	. . . . . . . . 9 |- ((x e. RR* /\ y e. RR*) -> (x(,)y) = {w e. RR* | (x < w /\ w < y)})
16	15	eqcomd 1889	. . . . . . . 8 |- ((x e. RR* /\ y e. RR*) -> {w e. RR* | (x < w /\ w < y)} = (x(,)y))
17	16	eqeq2d 1895	. . . . . . 7 |- ((x e. RR* /\ y e. RR*) -> (A = {w e. RR* | (x < w /\ w < y)} <-> A = (x(,)y)))
18	17	biimpd 170	. . . . . 6 |- ((x e. RR* /\ y e. RR*) -> (A = {w e. RR* | (x < w /\ w < y)} -> A = (x(,)y)))
19	18	imdistani 491	. . . . 5 |- (((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)}) -> ((x e. RR* /\ y e. RR*) /\ A = (x(,)y)))
20	19	2eximi 1388	. . . 4 |- (E.xE.y((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)}) -> E.xE.y((x e. RR* /\ y e. RR*) /\ A = (x(,)y)))
21		r2ex 2152	. . . 4 |- (E.x e. RR* E.y e. RR* A = (x(,)y) <-> E.xE.y((x e. RR* /\ y e. RR*) /\ A = (x(,)y)))
22	20, 21	sylibr 217	. . 3 |- (E.xE.y((x e. RR* /\ y e. RR*) /\ A = {w e. RR* | (x < w /\ w < y)}) -> E.x e. RR* E.y e. RR* A = (x(,)y))
23	14, 22	syl6 25	. 2 |- (A e. ran (,) -> (A e. ran (,) -> E.x e. RR* E.y e. RR* A = (x(,)y)))
24	23	pm2.43i 78	1 |- (A e. ran (,) -> E.x e. RR* E.y e. RR* A = (x(,)y))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106  {crab 2108   class class class wbr 3338  ran crn 3987  (class class class)co 4884  {copab2 4885  RR*cxr 6652   < clt 6653  (,)cioo 7524

This theorem is referenced by:  bsi3 14993  altretoplem2 14996

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-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-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-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-enr 6318  df-nr 6319  df-0r 6323  df-c 6392  df-r 6396  df-xr 6656  df-ioo 7528

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