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Theorem pinq 8809
 Description: The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pinq

Proof of Theorem pinq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pi 8765 . . . 4
2 opelxpi 4913 . . . 4
31, 2mpan2 654 . . 3
4 nlt1pi 8788 . . . . . 6
51elexi 2967 . . . . . . . 8
6 op2ndg 6363 . . . . . . . 8
75, 6mpan2 654 . . . . . . 7
87breq2d 4227 . . . . . 6
94, 8mtbiri 296 . . . . 5
109a1d 24 . . . 4
1110ralrimivw 2792 . . 3
12 breq1 4218 . . . . . 6
13 fveq2 5731 . . . . . . . 8
1413breq2d 4227 . . . . . . 7
1514notbid 287 . . . . . 6
1612, 15imbi12d 313 . . . . 5
1716ralbidv 2727 . . . 4
1817elrab 3094 . . 3
193, 11, 18sylanbrc 647 . 2
20 df-nq 8794 . 2
2119, 20syl6eleqr 2529 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wceq 1653   wcel 1726  wral 2707  crab 2711  cvv 2958  cop 3819   class class class wbr 4215   cxp 4879  cfv 5457  c2nd 6351  c1o 6720  cnpi 8724   clti 8727   ceq 8731  cnq 8732 This theorem is referenced by:  1nq  8810  archnq  8862  prlem934  8915 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-2nd 6353  df-1o 6727  df-ni 8754  df-lti 8757  df-nq 8794
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