Theorem ioo0 10402

Description: An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Assertion

Ref	Expression
ioo0	|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> B <_ A ) )

Proof of Theorem ioo0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iooval 10030	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = { x e. RR* | ( A < x /\ x < B ) } )
2	1	eqeq1d 2214	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> { x e. RR* | ( A < x /\ x < B ) } = (/) ) )
3		xrlttr 9917	. . . . . . . 8 |- ( ( A e. RR* /\ x e. RR* /\ B e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
4	3	3com23 1212	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
5	4	3expa 1206	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( ( A < x /\ x < B ) -> A < B ) )
6	5	rexlimdva 2623	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x < B ) -> A < B ) )
7		qbtwnxr 10400	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
8		qre 9746	. . . . . . . . . 10 |- ( x e. QQ -> x e. RR )
9	8	rexrd 8122	. . . . . . . . 9 |- ( x e. QQ -> x e. RR* )
10	9	anim1i 340	. . . . . . . 8 |- ( ( x e. QQ /\ ( A < x /\ x < B ) ) -> ( x e. RR* /\ ( A < x /\ x < B ) ) )
11	10	reximi2 2602	. . . . . . 7 |- ( E. x e. QQ ( A < x /\ x < B ) -> E. x e. RR* ( A < x /\ x < B ) )
12	7, 11	syl 14	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. RR* ( A < x /\ x < B ) )
13	12	3expia 1208	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. RR* ( A < x /\ x < B ) ) )
14	6, 13	impbid 129	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( E. x e. RR* ( A < x /\ x < B ) <-> A < B ) )
15	14	notbid 669	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( -. E. x e. RR* ( A < x /\ x < B ) <-> -. A < B ) )
16		rabeq0 3490	. . . . 5 |- ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> A. x e. RR* -. ( A < x /\ x < B ) )
17		ralnex 2494	. . . . 5 |- ( A. x e. RR* -. ( A < x /\ x < B ) <-> -. E. x e. RR* ( A < x /\ x < B ) )
18	16, 17	bitri 184	. . . 4 |- ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> -. E. x e. RR* ( A < x /\ x < B ) )
19	18	a1i 9	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> -. E. x e. RR* ( A < x /\ x < B ) ) )
20		xrlenlt 8137	. . . 4 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
21	20	ancoms 268	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
22	15, 19, 21	3bitr4d 220	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( { x e. RR* | ( A < x /\ x < B ) } = (/) <-> B <_ A ) )
23	2, 22	bitrd 188	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> B <_ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   {crab 2488   (/)c0 3460   class class class wbr 4044  (class class class)co 5944   RR*cxr 8106   < clt 8107   <_ cle 8108   QQcq 9740   (,)cioo 10010

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-ioo 10014

This theorem is referenced by: (None)