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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  |-  P  =  (PPoints `  I )
tpne.2  |-  ( ph  ->  I  e. Ig )
Assertion
Ref Expression
tpne  |-  ( ph  ->  P  =/=  (/) )
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem tpne
StepHypRef Expression
1 tpne.1 . . 3  |-  P  =  (PPoints `  I )
2 eqid 2286 . . 3  |-  (PLines `  I )  =  (PLines `  I )
3 tpne.2 . . 3  |-  ( ph  ->  I  e. Ig )
41, 2, 3tethpnc 25471 . 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 3559 . 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 936    = wceq 1625    e. wcel 1687    =/= wne 2449   A.wral 2546   E.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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