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Theorem ltbtwnnq 9785
Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltbtwnnq (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltbtwnnq
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 9733 . . . . 5 <Q ⊆ (Q × Q)
21brel 5158 . . . 4 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
32simprd 479 . . 3 (𝐴 <Q 𝐵𝐵Q)
4 ltexnq 9782 . . . 4 (𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑦(𝐴 +Q 𝑦) = 𝐵))
5 eleq1 2687 . . . . . . . . . 10 ((𝐴 +Q 𝑦) = 𝐵 → ((𝐴 +Q 𝑦) ∈ Q𝐵Q))
65biimparc 504 . . . . . . . . 9 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝐴 +Q 𝑦) ∈ Q)
7 addnqf 9755 . . . . . . . . . . 11 +Q :(Q × Q)⟶Q
87fdmi 6039 . . . . . . . . . 10 dom +Q = (Q × Q)
9 0nnq 9731 . . . . . . . . . 10 ¬ ∅ ∈ Q
108, 9ndmovrcl 6805 . . . . . . . . 9 ((𝐴 +Q 𝑦) ∈ Q → (𝐴Q𝑦Q))
116, 10syl 17 . . . . . . . 8 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝐴Q𝑦Q))
1211simprd 479 . . . . . . 7 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → 𝑦Q)
13 nsmallnq 9784 . . . . . . . 8 (𝑦Q → ∃𝑧 𝑧 <Q 𝑦)
1411simpld 475 . . . . . . . . . . . 12 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → 𝐴Q)
151brel 5158 . . . . . . . . . . . . 13 (𝑧 <Q 𝑦 → (𝑧Q𝑦Q))
1615simpld 475 . . . . . . . . . . . 12 (𝑧 <Q 𝑦𝑧Q)
17 ltaddnq 9781 . . . . . . . . . . . 12 ((𝐴Q𝑧Q) → 𝐴 <Q (𝐴 +Q 𝑧))
1814, 16, 17syl2an 494 . . . . . . . . . . 11 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → 𝐴 <Q (𝐴 +Q 𝑧))
19 ltanq 9778 . . . . . . . . . . . . . 14 (𝐴Q → (𝑧 <Q 𝑦 ↔ (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦)))
2019biimpa 501 . . . . . . . . . . . . 13 ((𝐴Q𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦))
2114, 20sylan 488 . . . . . . . . . . . 12 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦))
22 simplr 791 . . . . . . . . . . . 12 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑦) = 𝐵)
2321, 22breqtrd 4670 . . . . . . . . . . 11 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q 𝐵)
24 ovex 6663 . . . . . . . . . . . 12 (𝐴 +Q 𝑧) ∈ V
25 breq2 4648 . . . . . . . . . . . . 13 (𝑥 = (𝐴 +Q 𝑧) → (𝐴 <Q 𝑥𝐴 <Q (𝐴 +Q 𝑧)))
26 breq1 4647 . . . . . . . . . . . . 13 (𝑥 = (𝐴 +Q 𝑧) → (𝑥 <Q 𝐵 ↔ (𝐴 +Q 𝑧) <Q 𝐵))
2725, 26anbi12d 746 . . . . . . . . . . . 12 (𝑥 = (𝐴 +Q 𝑧) → ((𝐴 <Q 𝑥𝑥 <Q 𝐵) ↔ (𝐴 <Q (𝐴 +Q 𝑧) ∧ (𝐴 +Q 𝑧) <Q 𝐵)))
2824, 27spcev 3295 . . . . . . . . . . 11 ((𝐴 <Q (𝐴 +Q 𝑧) ∧ (𝐴 +Q 𝑧) <Q 𝐵) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
2918, 23, 28syl2anc 692 . . . . . . . . . 10 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
3029ex 450 . . . . . . . . 9 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝑧 <Q 𝑦 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3130exlimdv 1859 . . . . . . . 8 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (∃𝑧 𝑧 <Q 𝑦 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3213, 31syl5 34 . . . . . . 7 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝑦Q → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3312, 32mpd 15 . . . . . 6 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
3433ex 450 . . . . 5 (𝐵Q → ((𝐴 +Q 𝑦) = 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3534exlimdv 1859 . . . 4 (𝐵Q → (∃𝑦(𝐴 +Q 𝑦) = 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
364, 35sylbid 230 . . 3 (𝐵Q → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
373, 36mpcom 38 . 2 (𝐴 <Q 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
38 ltsonq 9776 . . . 4 <Q Or Q
3938, 1sotri 5511 . . 3 ((𝐴 <Q 𝑥𝑥 <Q 𝐵) → 𝐴 <Q 𝐵)
4039exlimiv 1856 . 2 (∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵) → 𝐴 <Q 𝐵)
4137, 40impbii 199 1 (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988   class class class wbr 4644   × cxp 5102  (class class class)co 6635  Qcnq 9659   +Q cplq 9662   <Q cltq 9665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-ni 9679  df-pli 9680  df-mi 9681  df-lti 9682  df-plpq 9715  df-mpq 9716  df-ltpq 9717  df-enq 9718  df-nq 9719  df-erq 9720  df-plq 9721  df-mq 9722  df-1nq 9723  df-rq 9724  df-ltnq 9725
This theorem is referenced by:  nqpr  9821  reclem2pr  9855
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