Theorem ioom 9269

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 8933	. . . . . . . 8 |- ( x e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) /\ ( A < x /\ x < B ) ) )
2	1	biimpi 118	. . . . . . 7 |- ( x e. ( A (,) B ) -> ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) /\ ( A < x /\ x < B ) ) )
3	2	simpld 110	. . . . . 6 |- ( x e. ( A (,) B ) -> ( A e. RR* /\ B e. RR* /\ x e. RR* ) )
4	3	simp1d 950	. . . . 5 |- ( x e. ( A (,) B ) -> A e. RR* )
5	3	simp3d 952	. . . . 5 |- ( x e. ( A (,) B ) -> x e. RR* )
6	3	simp2d 951	. . . . 5 |- ( x e. ( A (,) B ) -> B e. RR* )
7	2	simprd 112	. . . . . 6 |- ( x e. ( A (,) B ) -> ( A < x /\ x < B ) )
8	7	simpld 110	. . . . 5 |- ( x e. ( A (,) B ) -> A < x )
9	7	simprd 112	. . . . 5 |- ( x e. ( A (,) B ) -> x < B )
10	4, 5, 6, 8, 9	xrlttrd 8879	. . . 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 1740	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) -> A < B ) )
13		qbtwnxr 9266	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
14		df-rex 2354	. . . . 5 |- ( E. x e. QQ ( A < x /\ x < B ) <-> E. x ( x e. QQ /\ ( A < x /\ x < B ) ) )
15	13, 14	sylib 120	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x ( x e. QQ /\ ( A < x /\ x < B ) ) )
16		simpl1 941	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> A e. RR* )
17		simpl2 942	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> B e. RR* )
18		qre 8710	. . . . . . . . 9 |- ( x e. QQ -> x e. RR )
19	18	ad2antrl 473	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. RR )
20	19	rexrd 7168	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. RR* )
21		simprrl 505	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> A < x )
22		simprrr 506	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x < B )
23	1	biimpri 131	. . . . . . 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 1177	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. ( A (,) B ) )
25	24	ex 113	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> x e. ( A (,) B ) ) )
26	25	eximdv 1801	. . . 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 1140	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x x e. ( A (,) B ) ) )
29	12, 28	impbid 127	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) <-> A < B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   E.wex 1421   e. wcel 1433   E.wrex 2349   class class class wbr 3785  (class class class)co 5532   RRcr 6980   RR*cxr 7152   < clt 7153   QQcq 8704   (,)cioo 8911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-ioo 8915

This theorem is referenced by: (None)