Theorem ico0 10248

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 9906	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A [,) B ) = { x e. RR* | ( A <_ x /\ x < B ) } )
2	1	eqeq1d 2186	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> { x e. RR* | ( A <_ x /\ x < B ) } = (/) ) )
3		xrlelttr 9793	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
4	3	3com23 1209	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
5	4	3expa 1203	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A <_ x /\ x < B ) -> A < B ) )
6	5	rexlimdva 2594	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A <_ x /\ x < B ) -> A < B ) )
7		qbtwnxr 10244	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
8		qre 9614	. . . . . . . . . . . 12 |- ( x e. QQ -> x e. RR )
9	8	rexrd 7997	. . . . . . . . . . 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 1003	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> A e. RR* )
12		simpl 109	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> x e. RR* )
13		xrltle 9785	. . . . . . . . . . . . . . 15 |- ( ( A e. RR* /\ x e. RR* ) -> ( A < x -> A <_ x ) )
14	11, 12, 13	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( A < x -> A <_ x ) )
15	14	anim1d 336	. . . . . . . . . . . . 13 |- ( ( x e. RR* /\ ( A e. RR* /\ B e. RR* /\ A < B ) ) -> ( ( A < x /\ x < B ) -> ( A <_ x /\ x < B ) ) )
16	10, 15	anim12d 335	. . . . . . . . . . . 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 115	. . . . . . . . . . 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 276	. . . . . . . . 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 2576	. . . . . . 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 1205	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. RR* ( A <_ x /\ x < B ) ) )
24	6, 23	impbid 129	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A <_ x /\ x < B ) <-> A < B ) )
25	24	notbid 667	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. E. x e. RR* ( A <_ x /\ x < B ) <-> -. A < B ) )
26		rabeq0 3452	. . . . 5 |- ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> A. x e. RR* -. ( A <_ x /\ x < B ) )
27		ralnex 2465	. . . . 5 |- ( A. x e. RR* -. ( A <_ x /\ x < B ) <-> -. E. x e. RR* ( A <_ x /\ x < B ) )
28	26, 27	bitri 184	. . . 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 8012	. . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
31	30	ancoms 268	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
32	25, 29, 31	3bitr4d 220	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A <_ x /\ x < B ) } = (/) <-> B <_ A ) )
33	2, 32	bitrd 188	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,) B ) = (/) <-> B <_ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   (/)c0 3422   class class class wbr 4000  (class class class)co 5869   RR*cxr 7981   < clt 7982   <_ cle 7983   QQcq 9608   [,)cico 9877

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-ico 9881

This theorem is referenced by: (None)