Theorem qbtwnxr 10400

Description: The rational numbers are dense in RR*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
qbtwnxr	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem qbtwnxr

Step	Hyp	Ref	Expression
1		elxr 9898	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9898	. . . . 5 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		qbtwnre 10399	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
4	3	3expia 1208	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
5		simpl 109	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A e. RR )
6		peano2re 8208	. . . . . . . . . 10 |- ( A e. RR -> ( A + 1 ) e. RR )
7	6	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> ( A + 1 ) e. RR )
8		ltp1 8917	. . . . . . . . . 10 |- ( A e. RR -> A < ( A + 1 ) )
9	8	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A < ( A + 1 ) )
10		qbtwnre 10399	. . . . . . . . 9 |- ( ( A e. RR /\ ( A + 1 ) e. RR /\ A < ( A + 1 ) ) -> E. x e. QQ ( A < x /\ x < ( A + 1 ) ) )
11	5, 7, 9, 10	syl3anc 1250	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> E. x e. QQ ( A < x /\ x < ( A + 1 ) ) )
12		qre 9746	. . . . . . . . . . . . . 14 |- ( x e. QQ -> x e. RR )
13		ltpnf 9902	. . . . . . . . . . . . . 14 |- ( x e. RR -> x < +oo )
14	12, 13	syl 14	. . . . . . . . . . . . 13 |- ( x e. QQ -> x < +oo )
15	14	adantl 277	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> x < +oo )
16		simplr 528	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> B = +oo )
17	15, 16	breqtrrd 4072	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> x < B )
18	17	a1d 22	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> ( x < ( A + 1 ) -> x < B ) )
19	18	anim2d 337	. . . . . . . . 9 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> ( ( A < x /\ x < ( A + 1 ) ) -> ( A < x /\ x < B ) ) )
20	19	reximdva 2608	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> ( E. x e. QQ ( A < x /\ x < ( A + 1 ) ) -> E. x e. QQ ( A < x /\ x < B ) ) )
21	11, 20	mpd 13	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> E. x e. QQ ( A < x /\ x < B ) )
22	21	a1d 22	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
23		rexr 8118	. . . . . . 7 |- ( A e. RR -> A e. RR* )
24		breq2 4048	. . . . . . . . 9 |- ( B = -oo -> ( A < B <-> A < -oo ) )
25	24	adantl 277	. . . . . . . 8 |- ( ( A e. RR* /\ B = -oo ) -> ( A < B <-> A < -oo ) )
26		nltmnf 9910	. . . . . . . . . 10 |- ( A e. RR* -> -. A < -oo )
27	26	adantr 276	. . . . . . . . 9 |- ( ( A e. RR* /\ B = -oo ) -> -. A < -oo )
28	27	pm2.21d 620	. . . . . . . 8 |- ( ( A e. RR* /\ B = -oo ) -> ( A < -oo -> E. x e. QQ ( A < x /\ x < B ) ) )
29	25, 28	sylbid 150	. . . . . . 7 |- ( ( A e. RR* /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
30	23, 29	sylan 283	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
31	4, 22, 30	3jaodan 1319	. . . . 5 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
32	2, 31	sylan2b 287	. . . 4 |- ( ( A e. RR /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
33		breq1 4047	. . . . . 6 |- ( A = +oo -> ( A < B <-> +oo < B ) )
34	33	adantr 276	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
35		pnfnlt 9909	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
36	35	adantl 277	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
37	36	pm2.21d 620	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo < B -> E. x e. QQ ( A < x /\ x < B ) ) )
38	34, 37	sylbid 150	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
39		peano2rem 8339	. . . . . . . . . 10 |- ( B e. RR -> ( B - 1 ) e. RR )
40	39	adantl 277	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B - 1 ) e. RR )
41		simpr 110	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> B e. RR )
42		ltm1 8919	. . . . . . . . . 10 |- ( B e. RR -> ( B - 1 ) < B )
43	42	adantl 277	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B - 1 ) < B )
44		qbtwnre 10399	. . . . . . . . 9 |- ( ( ( B - 1 ) e. RR /\ B e. RR /\ ( B - 1 ) < B ) -> E. x e. QQ ( ( B - 1 ) < x /\ x < B ) )
45	40, 41, 43, 44	syl3anc 1250	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> E. x e. QQ ( ( B - 1 ) < x /\ x < B ) )
46		simpll 527	. . . . . . . . . . . 12 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> A = -oo )
47	12	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> x e. RR )
48		mnflt 9905	. . . . . . . . . . . . 13 |- ( x e. RR -> -oo < x )
49	47, 48	syl 14	. . . . . . . . . . . 12 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> -oo < x )
50	46, 49	eqbrtrd 4066	. . . . . . . . . . 11 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> A < x )
51	50	a1d 22	. . . . . . . . . 10 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> ( ( B - 1 ) < x -> A < x ) )
52	51	anim1d 336	. . . . . . . . 9 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> ( ( ( B - 1 ) < x /\ x < B ) -> ( A < x /\ x < B ) ) )
53	52	reximdva 2608	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> ( E. x e. QQ ( ( B - 1 ) < x /\ x < B ) -> E. x e. QQ ( A < x /\ x < B ) ) )
54	45, 53	mpd 13	. . . . . . 7 |- ( ( A = -oo /\ B e. RR ) -> E. x e. QQ ( A < x /\ x < B ) )
55	54	a1d 22	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
56		1re 8071	. . . . . . . . . 10 |- 1 e. RR
57		mnflt 9905	. . . . . . . . . 10 |- ( 1 e. RR -> -oo < 1 )
58	56, 57	ax-mp 5	. . . . . . . . 9 |- -oo < 1
59		breq1 4047	. . . . . . . . 9 |- ( A = -oo -> ( A < 1 <-> -oo < 1 ) )
60	58, 59	mpbiri 168	. . . . . . . 8 |- ( A = -oo -> A < 1 )
61		ltpnf 9902	. . . . . . . . . 10 |- ( 1 e. RR -> 1 < +oo )
62	56, 61	ax-mp 5	. . . . . . . . 9 |- 1 < +oo
63		breq2 4048	. . . . . . . . 9 |- ( B = +oo -> ( 1 < B <-> 1 < +oo ) )
64	62, 63	mpbiri 168	. . . . . . . 8 |- ( B = +oo -> 1 < B )
65		1z 9398	. . . . . . . . . 10 |- 1 e. ZZ
66		zq 9747	. . . . . . . . . 10 |- ( 1 e. ZZ -> 1 e. QQ )
67	65, 66	ax-mp 5	. . . . . . . . 9 |- 1 e. QQ
68		breq2 4048	. . . . . . . . . . 11 |- ( x = 1 -> ( A < x <-> A < 1 ) )
69		breq1 4047	. . . . . . . . . . 11 |- ( x = 1 -> ( x < B <-> 1 < B ) )
70	68, 69	anbi12d 473	. . . . . . . . . 10 |- ( x = 1 -> ( ( A < x /\ x < B ) <-> ( A < 1 /\ 1 < B ) ) )
71	70	rspcev 2877	. . . . . . . . 9 |- ( ( 1 e. QQ /\ ( A < 1 /\ 1 < B ) ) -> E. x e. QQ ( A < x /\ x < B ) )
72	67, 71	mpan 424	. . . . . . . 8 |- ( ( A < 1 /\ 1 < B ) -> E. x e. QQ ( A < x /\ x < B ) )
73	60, 64, 72	syl2an 289	. . . . . . 7 |- ( ( A = -oo /\ B = +oo ) -> E. x e. QQ ( A < x /\ x < B ) )
74	73	a1d 22	. . . . . 6 |- ( ( A = -oo /\ B = +oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
75		3mix3 1171	. . . . . . . 8 |- ( A = -oo -> ( A e. RR \/ A = +oo \/ A = -oo ) )
76	75, 1	sylibr 134	. . . . . . 7 |- ( A = -oo -> A e. RR* )
77	76, 29	sylan 283	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
78	55, 74, 77	3jaodan 1319	. . . . 5 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
79	2, 78	sylan2b 287	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
80	32, 38, 79	3jaoian 1318	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
81	1, 80	sylanb 284	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
82	81	3impia 1203	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 980   /\ w3a 981   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4044  (class class class)co 5944   RRcr 7924   1c1 7926   + caddc 7928   +oocpnf 8104   -oocmnf 8105   RR*cxr 8106   < clt 8107   - cmin 8243   ZZcz 9372   QQcq 9740

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-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

This theorem is referenced by:  ioo0  10402  ioom  10403  ico0  10404  ioc0  10405  blssps  14899  blss  14900  tgqioo  15027