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Theorem tpne 25476
 Description: The plane is not empty. Exercise 5 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 29-Apr-2016.)
Hypotheses
Ref Expression
tpne.1 PPoints
tpne.2 Ig
Assertion
Ref Expression
tpne
Dummy variables are mutually distinct and distinct from all other variables.

Proof of Theorem tpne
StepHypRef Expression
1 tpne.1 . . 3 PPoints
2 eqid 2286 . . 3 PLines PLines
3 tpne.2 . . 3 Ig
41, 2, 3tethpnc 25471 . 2 PLines
5 rexn0 3559 . 2 PLines
64, 5syl 17 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wa 360   w3a 936   wceq 1625   wcel 1687   wne 2449  wral 2546  wrex 2547  c0 3458  cfv 5223  PPointscpoints 25457  PLinescplines 25459  Igcig 25461 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pr 4215  ax-un 4513 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-reu 2553  df-rab 2555  df-v 2793  df-sbc 2995  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-xp 4696  df-cnv 4698  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fv 5231  df-ig2 25462
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