Theorem ioom 10480

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 10106	. . . . . . . 8 |- ( x e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) /\ ( A < x /\ x < B ) ) )
2	1	biimpi 120	. . . . . . 7 |- ( x e. ( A (,) B ) -> ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) /\ ( A < x /\ x < B ) ) )
3	2	simpld 112	. . . . . 6 |- ( x e. ( A (,) B ) -> ( A e. RR* /\ B e. RR* /\ x e. RR* ) )
4	3	simp1d 1033	. . . . 5 |- ( x e. ( A (,) B ) -> A e. RR* )
5	3	simp3d 1035	. . . . 5 |- ( x e. ( A (,) B ) -> x e. RR* )
6	3	simp2d 1034	. . . . 5 |- ( x e. ( A (,) B ) -> B e. RR* )
7	2	simprd 114	. . . . . 6 |- ( x e. ( A (,) B ) -> ( A < x /\ x < B ) )
8	7	simpld 112	. . . . 5 |- ( x e. ( A (,) B ) -> A < x )
9	7	simprd 114	. . . . 5 |- ( x e. ( A (,) B ) -> x < B )
10	4, 5, 6, 8, 9	xrlttrd 10005	. . . 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 1865	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) -> A < B ) )
13		qbtwnxr 10477	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
14		df-rex 2514	. . . . 5 |- ( E. x e. QQ ( A < x /\ x < B ) <-> E. x ( x e. QQ /\ ( A < x /\ x < B ) ) )
15	13, 14	sylib 122	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x ( x e. QQ /\ ( A < x /\ x < B ) ) )
16		simpl1 1024	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> A e. RR* )
17		simpl2 1025	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> B e. RR* )
18		qre 9820	. . . . . . . . 9 |- ( x e. QQ -> x e. RR )
19	18	ad2antrl 490	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. RR )
20	19	rexrd 8196	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. RR* )
21		simprrl 539	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> A < x )
22		simprrr 540	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x < B )
23	1	biimpri 133	. . . . . . 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 1279	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ A < B ) /\ ( x e. QQ /\ ( A < x /\ x < B ) ) ) -> x e. ( A (,) B ) )
25	24	ex 115	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> x e. ( A (,) B ) ) )
26	25	eximdv 1926	. . . 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 1229	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x x e. ( A (,) B ) ) )
29	12, 28	impbid 129	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x x e. ( A (,) B ) <-> A < B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   E.wex 1538   e. wcel 2200   E.wrex 2509   class class class wbr 4083  (class class class)co 6001   RRcr 7998   RR*cxr 8180   < clt 8181   QQcq 9814   (,)cioo 10084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-ioo 10088

This theorem is referenced by:  tgioo  15228