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Theorem tpne 25407
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 2256 . . 3  |-  (PLines `  I )  =  (PLines `  I )
3 tpne.2 . . 3  |-  ( ph  ->  I  e. Ig )
41, 2, 3tethpnc 25402 . 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 3498 . 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 2419   A.wral 2516   E.wrex 2517   (/)c0 3397   ` cfv 4638  PPointscpoints 25388  PLinescplines 25390  Igcig 25392
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fv 4654  df-ig2 25393
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