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Theorem elpell1234qr 26928
 Description: Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
elpell1234qr NN Pell1234QR
Distinct variable groups:   ,,   ,,

Proof of Theorem elpell1234qr
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pell1234qrval 26927 . . 3 NN Pell1234QR
21eleq2d 2505 . 2 NN Pell1234QR
3 eqeq1 2444 . . . . 5
43anbi1d 687 . . . 4
542rexbidv 2750 . . 3
65elrab 3094 . 2
72, 6syl6bb 254 1 NN Pell1234QR
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wrex 2708  crab 2711   cdif 3319  cfv 5457  (class class class)co 6084  cr 8994  c1 8996   caddc 8998   cmul 9000   cmin 9296  cn 10005  c2 10054  cz 10287  cexp 11387  csqr 12043  ◻NNcsquarenn 26913  Pell1234QRcpell1234qr 26915 This theorem is referenced by:  pell1234qrre  26929  pell1234qrne0  26930  pell1234qrreccl  26931  pell1234qrmulcl  26932  pell14qrss1234  26933  pell1234qrdich  26938 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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-cnex 9051  ax-resscn 9052 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-pell1234qr 26921
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