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Theorem elpell1qr 26901
 Description: Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
elpell1qr NN Pell1QR
Distinct variable groups:   ,,   ,,

Proof of Theorem elpell1qr
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pell1qrval 26900 . . 3 NN Pell1QR
21eleq2d 2502 . 2 NN Pell1QR
3 eqeq1 2441 . . . . 5
43anbi1d 686 . . . 4
542rexbidv 2740 . . 3
65elrab 3084 . 2
72, 6syl6bb 253 1 NN Pell1QR
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wrex 2698  crab 2701   cdif 3309  cfv 5446  (class class class)co 6073  cr 8981  c1 8983   caddc 8985   cmul 8987   cmin 9283  cn 9992  c2 10041  cn0 10213  cexp 11374  csqr 12030  ◻NNcsquarenn 26890  Pell1QRcpell1qr 26891 This theorem is referenced by:  pell1qrss14  26922  pell14qrdich  26923  pell1qrge1  26924  pell1qr1  26925  pell1qrgap  26928  pellqrexplicit  26931 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-cnex 9038  ax-resscn 9039 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-pell1qr 26896
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