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Theorem tpne 25241
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  |-  P  =  (PPoints `  I )
tpne.2  |-  ( ph  ->  I  e. Ig )
Assertion
Ref Expression
tpne  |-  ( ph  ->  P  =/=  (/) )

Proof of Theorem tpne
StepHypRef Expression
1 tpne.1 . . 3  |-  P  =  (PPoints `  I )
2 eqid 2253 . . 3  |-  (PLines `  I )  =  (PLines `  I )
3 tpne.2 . . 3  |-  ( ph  ->  I  e. Ig )
41, 2, 3tethpnc 25236 . 2  |-  ( ph  ->  E. x  e.  P  E. y  e.  P  E. z  e.  P  ( ( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z )  /\  A. w  e.  (PLines `  I
)  -.  ( x  e.  w  /\  y  e.  w  /\  z  e.  w ) ) )
5 rexn0 3462 . 2  |-  ( E. x  e.  P  E. y  e.  P  E. z  e.  P  (
( x  =/=  y  /\  y  =/=  z  /\  x  =/=  z
)  /\  A. w  e.  (PLines `  I )  -.  ( x  e.  w  /\  y  e.  w  /\  z  e.  w
) )  ->  P  =/=  (/) )
64, 5syl 17 1  |-  ( ph  ->  P  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510   (/)c0 3362   ` cfv 4592  PPointscpoints 25222  PLinescplines 25224  Igcig 25226
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ig2 25227
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