Theorem ioom 10038

Description: An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.)

Assertion

Ref	Expression
ioom	|- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) <-> A < B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem ioom

Step	Hyp	Ref	Expression
1		elioo3g 9693	. . . . . . . 8 |- ( x e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) /\ ( A < x /\ x < B ) ) )
2	1	biimpi 119	. . . . . . 7 |- ( x e. ( A (,) B ) -> ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) /\ ( A < x /\ x < B ) ) )
3	2	simpld 111	. . . . . 6 |- ( x e. ( A (,) B ) -> ( A e. RR* /\ B e. RR* /\ x e. RR* ) )
4	3	simp1d 993	. . . . 5 |- ( x e. ( A (,) B ) -> A e. RR* )
5	3	simp3d 995	. . . . 5 |- ( x e. ( A (,) B ) -> x e. RR* )
6	3	simp2d 994	. . . . 5 |- ( x e. ( A (,) B ) -> B e. RR* )
7	2	simprd 113	. . . . . 6 |- ( x e. ( A (,) B ) -> ( A < x /\ x < B ) )
8	7	simpld 111	. . . . 5 |- ( x e. ( A (,) B ) -> A < x )
9	7	simprd 113	. . . . 5 |- ( x e. ( A (,) B ) -> x < B )
10	4, 5, 6, 8, 9	xrlttrd 9592	. . . 4 |- ( x e. ( A (,) B ) -> A < B )
11	10	a1i 9	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A (,) B ) -> A < B ) )
12	11	exlimdv 1791	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) -> A < B ) )
13		qbtwnxr 10035	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
14		df-rex 2422	. . . . 5 |- ( E. x e. QQ ( A < x /\ x < B ) <-> E. x ( x e. QQ /\ ( A < x /\ x < B ) ) )
15	13, 14	sylib 121	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x ( x e. QQ /\ ( A < x /\ x < B ) ) )
16		simpl1 984	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> A e. RR* )
17		simpl2 985	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> B e. RR* )
18		qre 9417	. . . . . . . . 9 |- ( x e. QQ -> x e. RR )
19	18	ad2antrl 481	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. RR )
20	19	rexrd 7815	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. RR* )
21		simprrl 528	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> A < x )
22		simprrr 529	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x < B )
23	1	biimpri 132	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) /\ ( A < x /\ x < B ) ) -> x e. ( A (,) B ) )
24	16, 17, 20, 21, 22, 23	syl32anc 1224	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. ( A (,) B ) )
25	24	ex 114	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> x e. ( A (,) B ) ) )
26	25	eximdv 1852	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( E. x ( x e. QQ /\ ( A < x /\ x < B ) ) -> E. x x e. ( A (,) B ) ) )
27	15, 26	mpd 13	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x x e. ( A (,) B ) )
28	27	3expia 1183	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x x e. ( A (,) B ) ) )
29	12, 28	impbid 128	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) <-> A < B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   E.wex 1468   e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   RRcr 7619   RR*cxr 7799   < clt 7800   QQcq 9411   (,)cioo 9671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ioo 9675

This theorem is referenced by:  tgioo  12715