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Theorem ltbtwnnq 10053
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 10001 . . . . 5 <Q ⊆ (Q × Q)
21brel 5336 . . . 4 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
32simprd 489 . . 3 (𝐴 <Q 𝐵𝐵Q)
4 ltexnq 10050 . . . 4 (𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑦(𝐴 +Q 𝑦) = 𝐵))
5 eleq1 2832 . . . . . . . . . 10 ((𝐴 +Q 𝑦) = 𝐵 → ((𝐴 +Q 𝑦) ∈ Q𝐵Q))
65biimparc 471 . . . . . . . . 9 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝐴 +Q 𝑦) ∈ Q)
7 addnqf 10023 . . . . . . . . . . 11 +Q :(Q × Q)⟶Q
87fdmi 6233 . . . . . . . . . 10 dom +Q = (Q × Q)
9 0nnq 9999 . . . . . . . . . 10 ¬ ∅ ∈ Q
108, 9ndmovrcl 7018 . . . . . . . . 9 ((𝐴 +Q 𝑦) ∈ Q → (𝐴Q𝑦Q))
116, 10syl 17 . . . . . . . 8 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝐴Q𝑦Q))
1211simprd 489 . . . . . . 7 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → 𝑦Q)
13 nsmallnq 10052 . . . . . . . 8 (𝑦Q → ∃𝑧 𝑧 <Q 𝑦)
1411simpld 488 . . . . . . . . . . . 12 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → 𝐴Q)
151brel 5336 . . . . . . . . . . . . 13 (𝑧 <Q 𝑦 → (𝑧Q𝑦Q))
1615simpld 488 . . . . . . . . . . . 12 (𝑧 <Q 𝑦𝑧Q)
17 ltaddnq 10049 . . . . . . . . . . . 12 ((𝐴Q𝑧Q) → 𝐴 <Q (𝐴 +Q 𝑧))
1814, 16, 17syl2an 589 . . . . . . . . . . 11 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → 𝐴 <Q (𝐴 +Q 𝑧))
19 ltanq 10046 . . . . . . . . . . . . . 14 (𝐴Q → (𝑧 <Q 𝑦 ↔ (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦)))
2019biimpa 468 . . . . . . . . . . . . 13 ((𝐴Q𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦))
2114, 20sylan 575 . . . . . . . . . . . 12 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q (𝐴 +Q 𝑦))
22 simplr 785 . . . . . . . . . . . 12 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑦) = 𝐵)
2321, 22breqtrd 4835 . . . . . . . . . . 11 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → (𝐴 +Q 𝑧) <Q 𝐵)
24 ovex 6874 . . . . . . . . . . . 12 (𝐴 +Q 𝑧) ∈ V
25 breq2 4813 . . . . . . . . . . . . 13 (𝑥 = (𝐴 +Q 𝑧) → (𝐴 <Q 𝑥𝐴 <Q (𝐴 +Q 𝑧)))
26 breq1 4812 . . . . . . . . . . . . 13 (𝑥 = (𝐴 +Q 𝑧) → (𝑥 <Q 𝐵 ↔ (𝐴 +Q 𝑧) <Q 𝐵))
2725, 26anbi12d 624 . . . . . . . . . . . 12 (𝑥 = (𝐴 +Q 𝑧) → ((𝐴 <Q 𝑥𝑥 <Q 𝐵) ↔ (𝐴 <Q (𝐴 +Q 𝑧) ∧ (𝐴 +Q 𝑧) <Q 𝐵)))
2824, 27spcev 3452 . . . . . . . . . . 11 ((𝐴 <Q (𝐴 +Q 𝑧) ∧ (𝐴 +Q 𝑧) <Q 𝐵) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
2918, 23, 28syl2anc 579 . . . . . . . . . 10 (((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) ∧ 𝑧 <Q 𝑦) → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
3029ex 401 . . . . . . . . 9 ((𝐵Q ∧ (𝐴 +Q 𝑦) = 𝐵) → (𝑧 <Q 𝑦 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3130exlimdv 2028 . . . . . . . 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 401 . . . . 5 (𝐵Q → ((𝐴 +Q 𝑦) = 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
3534exlimdv 2028 . . . 4 (𝐵Q → (∃𝑦(𝐴 +Q 𝑦) = 𝐵 → ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵)))
364, 35sylbid 231 . . 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 10044 . . . 4 <Q Or Q
3938, 1sotri 5706 . . 3 ((𝐴 <Q 𝑥𝑥 <Q 𝐵) → 𝐴 <Q 𝐵)
4039exlimiv 2025 . 2 (∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵) → 𝐴 <Q 𝐵)
4137, 40impbii 200 1 (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155   class class class wbr 4809   × cxp 5275  (class class class)co 6842  Qcnq 9927   +Q cplq 9930   <Q cltq 9933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-omul 7769  df-er 7947  df-ni 9947  df-pli 9948  df-mi 9949  df-lti 9950  df-plpq 9983  df-mpq 9984  df-ltpq 9985  df-enq 9986  df-nq 9987  df-erq 9988  df-plq 9989  df-mq 9990  df-1nq 9991  df-rq 9992  df-ltnq 9993
This theorem is referenced by:  nqpr  10089  reclem2pr  10123
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