Theorem bsi 10495

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 1481	. . . . . . . . 9 |- (z = A -> (z = {w e. RR* | (x < w /\ w < y)} <-> A = {w e. RR* | (x < w /\ w < y)}))
2	1	anbi2d 616	. . . . . . . 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 1281	. . . . . . 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 1888	. . . . . 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 153	. . . . 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 1581	. . . . 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 206	. . . 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 6361	. . . . . . 7 |- (,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
9	8	rneqi 3340	. . . . . 6 |- ran (,) = ran {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
10	9	eleq2i 1538	. . . . 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 4004	. . . . . 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 1538	. . . . 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	bitr 173	. . . 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 206	. . 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		ioovalt 6366	. . . . . . . . 9 |- ((x e. RR* /\ y e. RR*) -> (x(,)y) = {w e. RR* | (x < w /\ w < y)})
16	15	eqcomd 1480	. . . . . . . 8 |- ((x e. RR* /\ y e. RR*) -> {w e. RR* | (x < w /\ w < y)} = (x(,)y))
17	16	eqeq2d 1486	. . . . . . 7 |- ((x e. RR* /\ y e. RR*) -> (A = {w e. RR* | (x < w /\ w < y)} <-> A = (x(,)y)))
18	17	biimpd 153	. . . . . 6 |- ((x e. RR* /\ y e. RR*) -> (A = {w e. RR* | (x < w /\ w < y)} -> A = (x(,)y)))
19	18	imdistani 443	. . . . 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	19.22i2 1041	. . . 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 1691	. . . 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 200	. . 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 22	. 2 |- (A e. ran (,) -> (A e. ran (,) -> E.x e. RR* E.y e. RR* A = (x(,)y)))
24	23	pm2.43i 64	1 |- (A e. ran (,) -> E.x e. RR* E.y e. RR* A = (x(,)y))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  {crab 1648   class class class wbr 2619  ran crn 3171  (class class class)co 3963  {copab2 3964  RR*cxr 5485   < clt 5486  (,)cioo 6357

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-enr 5166  df-nr 5167  df-0r 5171  df-c 5240  df-r 5244  df-xr 5489  df-ioo 6361

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