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Theorem ltbtwnnq 11018
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 10966 . . . . 5 <Q ⊆ (Q × Q)
21brel 5750 . . . 4 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
32simprd 495 . . 3 (𝐴 <Q 𝐵𝐵Q)
4 ltexnq 11015 . . . 4 (𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑦(𝐴 +Q 𝑦) = 𝐵))
5 eleq1 2829 . . . . . . . . . 10 ((𝐴 +Q 𝑦) = 𝐵 → ((𝐴 +Q 𝑦) ∈ Q𝐵Q))
65biimparc 479 . . . . . . . . 9 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝐴 +Q 𝑦) ∈ Q)
7 addnqf 10988 . . . . . . . . . . 11 +Q :(Q × Q)⟶Q
87fdmi 6747 . . . . . . . . . 10 dom +Q = (Q × Q)
9 0nnq 10964 . . . . . . . . . 10 ¬ ∅ ∈ Q
108, 9ndmovrcl 7619 . . . . . . . . 9 ((𝐴 +Q 𝑦) ∈ Q → (𝐴Q𝑦Q))
116, 10syl 17 . . . . . . . 8 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝐴Q𝑦Q))
1211simprd 495 . . . . . . 7 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → 𝑦Q)
13 nsmallnq 11017 . . . . . . . 8 (𝑦Q → ∃𝑧 𝑧 <Q 𝑦)
1411simpld 494 . . . . . . . . . . . 12 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → 𝐴Q)
151brel 5750 . . . . . . . . . . . . 13 (𝑧 <Q 𝑦 → (𝑧Q𝑦Q))
1615simpld 494 . . . . . . . . . . . 12 (𝑧 <Q 𝑦𝑧Q)
17 ltaddnq 11014 . . . . . . . . . . . 12 ((𝐴Q𝑧Q) → 𝐴 <Q (𝐴 +Q 𝑧))
1814, 16, 17syl2an 596 . . . . . . . . . . 11 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → 𝐴 <Q (𝐴 +Q 𝑧))
19 ltanq 11011 . . . . . . . . . . . . . 14 (𝐴Q → (𝑧 <Q 𝑦 ↔ (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦)))
2019biimpa 476 . . . . . . . . . . . . 13 ((𝐴Q𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦))
2114, 20sylan 580 . . . . . . . . . . . 12 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦))
22 simplr 769 . . . . . . . . . . . 12 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑦) = 𝐵)
2321, 22breqtrd 5169 . . . . . . . . . . 11 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q 𝐵)
24 ovex 7464 . . . . . . . . . . . 12 (𝐴 +Q 𝑧) ∈ V
25 breq2 5147 . . . . . . . . . . . . 13 (𝑥 = (𝐴 +Q 𝑧) → (𝐴 <Q 𝑥𝐴 <Q (𝐴 +Q 𝑧)))
26 breq1 5146 . . . . . . . . . . . . 13 (𝑥 = (𝐴 +Q 𝑧) → (𝑥 <Q 𝐵 ↔ (𝐴 +Q 𝑧) <Q 𝐵))
2725, 26anbi12d 632 . . . . . . . . . . . 12 (𝑥 = (𝐴 +Q 𝑧) → ((𝐴 <Q 𝑥𝑥 <Q 𝐵) ↔ (𝐴 <Q (𝐴 +Q 𝑧) ∧ (𝐴 +Q 𝑧) <Q 𝐵)))
2824, 27spcev 3606 . . . . . . . . . . 11 ((𝐴 <Q (𝐴 +Q 𝑧) ∧ (𝐴 +Q 𝑧) <Q 𝐵) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
2918, 23, 28syl2anc 584 . . . . . . . . . 10 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
3029ex 412 . . . . . . . . 9 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝑧 <Q 𝑦 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3130exlimdv 1933 . . . . . . . 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 412 . . . . 5 (𝐵Q → ((𝐴 +Q 𝑦) = 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3534exlimdv 1933 . . . 4 (𝐵Q → (∃𝑦(𝐴 +Q 𝑦) = 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
364, 35sylbid 240 . . 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 11009 . . . 4 <Q Or Q
3938, 1sotri 6147 . . 3 ((𝐴 <Q 𝑥𝑥 <Q 𝐵) → 𝐴 <Q 𝐵)
4039exlimiv 1930 . 2 (∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵) → 𝐴 <Q 𝐵)
4137, 40impbii 209 1 (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108   class class class wbr 5143   × cxp 5683  (class class class)co 7431  Qcnq 10892   +Q cplq 10895   <Q cltq 10898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-plpq 10948  df-mpq 10949  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-plq 10954  df-mq 10955  df-1nq 10956  df-rq 10957  df-ltnq 10958
This theorem is referenced by:  nqpr  11054  reclem2pr  11088
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