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Theorem elpell14qr 26912
 Description: Membership in the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
elpell14qr NN Pell14QR
Distinct variable groups:   ,,   ,,

Proof of Theorem elpell14qr
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pell14qrval 26911 . . 3 NN Pell14QR
21eleq2d 2503 . 2 NN Pell14QR
3 eqeq1 2442 . . . . 5
43anbi1d 686 . . . 4
542rexbidv 2748 . . 3
65elrab 3092 . 2
72, 6syl6bb 253 1 NN Pell14QR
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wrex 2706  crab 2709   cdif 3317  cfv 5454  (class class class)co 6081  cr 8989  c1 8991   caddc 8993   cmul 8995   cmin 9291  cn 10000  c2 10049  cn0 10221  cz 10282  cexp 11382  csqr 12038  ◻NNcsquarenn 26899  Pell14QRcpell14qr 26902 This theorem is referenced by:  pell14qrss1234  26919  pell14qrgt0  26922  pell1234qrdich  26924  pell1qrss14  26931  pell14qrdich  26932  rmxycomplete  26980 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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-cnex 9046  ax-resscn 9047 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-pell14qr 26906
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