Theorem ico0 10218

Description: An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Assertion

Ref	Expression
ico0	|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) )

Proof of Theorem ico0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icoval 9876	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) = { x e. RR* | ( A <_ x /\ x < B ) } )
2	1	eqeq1d 2179	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> { x e. RR* | ( A <_ x /\ x < B ) } = (/) ) )
3		xrlelttr 9763	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
4	3	3com23 1204	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
5	4	3expa 1198	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
6	5	rexlimdva 2587	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A <_ x /\ x < B ) -> A < B ) )
7		qbtwnxr 10214	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
8		qre 9584	. . . . . . . . . . . 12 |- ( x e. QQ -> x e. RR )
9	8	rexrd 7969	. . . . . . . . . . 11 |- ( x e. QQ -> x e. RR* )
10	9	a1i 9	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( x e. QQ -> x e. RR* ) )
11		simpr1 998	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> A e. RR* )
12		simpl 108	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> x e. RR* )
13		xrltle 9755	. . . . . . . . . . . . . . 15 |- ( ( A e. RR* /\ x e. RR* ) -> ( A < x -> A <_ x ) )
14	11, 12, 13	syl2anc 409	. . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( A < x -> A <_ x ) )
15	14	anim1d 334	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( ( A < x /\ x < B ) -> ( A <_ x /\ x < B ) ) )
16	10, 15	anim12d 333	. . . . . . . . . . . 12 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) )
17	16	ex 114	. . . . . . . . . . 11 |- ( x e. RR* -> ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) ) )
18	9, 17	syl 14	. . . . . . . . . 10 |- ( x e. QQ -> ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) ) )
19	18	adantr 274	. . . . . . . . 9 |- ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) ) )
20	19	pm2.43b 52	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A <_ x /\ x < B ) ) ) )
21	20	reximdv2 2569	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( E. x e. QQ ( A < x /\ x < B ) -> E. x e. RR* ( A <_ x /\ x < B ) ) )
22	7, 21	mpd 13	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. RR* ( A <_ x /\ x < B ) )
23	22	3expia 1200	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. RR* ( A <_ x /\ x < B ) ) )
24	6, 23	impbid 128	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A <_ x /\ x < B ) <-> A < B ) )
25	24	notbid 662	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. E. x e. RR* ( A <_ x /\ x < B ) <-> -. A < B ) )
26		rabeq0 3444	. . . . 5 |- ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> A. x e. RR* -. ( A <_ x /\ x < B ) )
27		ralnex 2458	. . . . 5 |- ( A. x e. RR* -. ( A <_ x /\ x < B ) <-> -. E. x e. RR* ( A <_ x /\ x < B ) )
28	26, 27	bitri 183	. . . 4 |- ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> -. E. x e. RR* ( A <_ x /\ x < B ) )
29	28	a1i 9	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> -. E. x e. RR* ( A <_ x /\ x < B ) ) )
30		xrlenlt 7984	. . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
31	30	ancoms 266	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
32	25, 29, 31	3bitr4d 219	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> B <_ A ) )
33	2, 32	bitrd 187	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   (/)c0 3414   class class class wbr 3989  (class class class)co 5853   RR*cxr 7953   < clt 7954   <_ cle 7955   QQcq 9578   [,)cico 9847

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851

This theorem is referenced by: (None)