Theorem bsi 10481

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 1484	. . . . . . . . 9 |- (z = A -> (z = {w e. RR* | (x < w /\ w < y)} <-> A = {w e. RR* | (x < w /\ w < y)}))
2	1	anbi2d 618	. . . . . . . 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 1283	. . . . . . 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 1891	. . . . . 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 1584	. . . . 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 6362	. . . . . . 7 |- (,) = {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
9	8	rneqi 3346	. . . . . 6 |- ran (,) = ran {<.<.x, y>., z>. | ((x e. RR* /\ y e. RR*) /\ z = {w e. RR* | (x < w /\ w < y)})}
10	9	eleq2i 1541	. . . . 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 4010	. . . . . 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 1541	. . . . 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 6367	. . . . . . . . 9 |- ((x e. RR* /\ y e. RR*) -> (x(,)y) = {w e. RR* | (x < w /\ w < y)})
16	15	eqcomd 1483	. . . . . . . 8 |- ((x e. RR* /\ y e. RR*) -> {w e. RR* | (x < w /\ w < y)} = (x(,)y))
17	16	eqeq2d 1489	. . . . . . 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 445	. . . . 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 1043	. . . 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 1694	. . . 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 958   e. wcel 960  E.wex 982  {cab 1466  E.wrex 1649  {crab 1651   class class class wbr 2624  ran crn 3177  (class class class)co 3969  {copab2 3970  RR*cxr 5497   < clt 5498  (,)cioo 6358

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-enr 5178  df-nr 5179  df-0r 5183  df-c 5252  df-r 5256  df-xr 5501  df-ioo 6362

Copyright terms: Public domain