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Theorem ltbtwnnq 9652
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 9600 . . . . 5 <Q ⊆ (Q × Q)
21brel 5076 . . . 4 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
32simprd 477 . . 3 (𝐴 <Q 𝐵𝐵Q)
4 ltexnq 9649 . . . 4 (𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑦(𝐴 +Q 𝑦) = 𝐵))
5 eleq1 2671 . . . . . . . . . 10 ((𝐴 +Q 𝑦) = 𝐵 → ((𝐴 +Q 𝑦) ∈ Q𝐵Q))
65biimparc 502 . . . . . . . . 9 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝐴 +Q 𝑦) ∈ Q)
7 addnqf 9622 . . . . . . . . . . 11 +Q :(Q × Q)⟶Q
87fdmi 5947 . . . . . . . . . 10 dom +Q = (Q × Q)
9 0nnq 9598 . . . . . . . . . 10 ¬ ∅ ∈ Q
108, 9ndmovrcl 6691 . . . . . . . . 9 ((𝐴 +Q 𝑦) ∈ Q → (𝐴Q𝑦Q))
116, 10syl 17 . . . . . . . 8 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝐴Q𝑦Q))
1211simprd 477 . . . . . . 7 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → 𝑦Q)
13 nsmallnq 9651 . . . . . . . 8 (𝑦Q → ∃𝑧 𝑧 <Q 𝑦)
1411simpld 473 . . . . . . . . . . . 12 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → 𝐴Q)
151brel 5076 . . . . . . . . . . . . 13 (𝑧 <Q 𝑦 → (𝑧Q𝑦Q))
1615simpld 473 . . . . . . . . . . . 12 (𝑧 <Q 𝑦𝑧Q)
17 ltaddnq 9648 . . . . . . . . . . . 12 ((𝐴Q𝑧Q) → 𝐴 <Q (𝐴 +Q 𝑧))
1814, 16, 17syl2an 492 . . . . . . . . . . 11 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → 𝐴 <Q (𝐴 +Q 𝑧))
19 ltanq 9645 . . . . . . . . . . . . . 14 (𝐴Q → (𝑧 <Q 𝑦 ↔ (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦)))
2019biimpa 499 . . . . . . . . . . . . 13 ((𝐴Q𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦))
2114, 20sylan 486 . . . . . . . . . . . 12 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦))
22 simplr 787 . . . . . . . . . . . 12 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑦) = 𝐵)
2321, 22breqtrd 4599 . . . . . . . . . . 11 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q 𝐵)
24 ovex 6551 . . . . . . . . . . . 12 (𝐴 +Q 𝑧) ∈ V
25 breq2 4577 . . . . . . . . . . . . 13 (𝑥 = (𝐴 +Q 𝑧) → (𝐴 <Q 𝑥𝐴 <Q (𝐴 +Q 𝑧)))
26 breq1 4576 . . . . . . . . . . . . 13 (𝑥 = (𝐴 +Q 𝑧) → (𝑥 <Q 𝐵 ↔ (𝐴 +Q 𝑧) <Q 𝐵))
2725, 26anbi12d 742 . . . . . . . . . . . 12 (𝑥 = (𝐴 +Q 𝑧) → ((𝐴 <Q 𝑥𝑥 <Q 𝐵) ↔ (𝐴 <Q (𝐴 +Q 𝑧) ∧ (𝐴 +Q 𝑧) <Q 𝐵)))
2824, 27spcev 3268 . . . . . . . . . . 11 ((𝐴 <Q (𝐴 +Q 𝑧) ∧ (𝐴 +Q 𝑧) <Q 𝐵) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
2918, 23, 28syl2anc 690 . . . . . . . . . 10 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
3029ex 448 . . . . . . . . 9 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝑧 <Q 𝑦 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3130exlimdv 1846 . . . . . . . 8 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (∃𝑧 𝑧 <Q 𝑦 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3213, 31syl5 33 . . . . . . 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 448 . . . . 5 (𝐵Q → ((𝐴 +Q 𝑦) = 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3534exlimdv 1846 . . . 4 (𝐵Q → (∃𝑦(𝐴 +Q 𝑦) = 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
364, 35sylbid 228 . . 3 (𝐵Q → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
373, 36mpcom 37 . 2 (𝐴 <Q 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
38 ltsonq 9643 . . . 4 <Q Or Q
3938, 1sotri 5425 . . 3 ((𝐴 <Q 𝑥𝑥 <Q 𝐵) → 𝐴 <Q 𝐵)
4039exlimiv 1843 . 2 (∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵) → 𝐴 <Q 𝐵)
4137, 40impbii 197 1 (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1975   class class class wbr 4573   × cxp 5022  (class class class)co 6523  Qcnq 9526   +Q cplq 9529   <Q cltq 9532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-omul 7425  df-er 7602  df-ni 9546  df-pli 9547  df-mi 9548  df-lti 9549  df-plpq 9582  df-mpq 9583  df-ltpq 9584  df-enq 9585  df-nq 9586  df-erq 9587  df-plq 9588  df-mq 9589  df-1nq 9590  df-rq 9591  df-ltnq 9592
This theorem is referenced by:  nqpr  9688  reclem2pr  9722
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