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Theorem tpne 26075
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
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tpne.1 . . 3  |-  P  =  (PPoints `  I )
2 eqid 2283 . . 3  |-  (PLines `  I )  =  (PLines `  I )
3 tpne.2 . . 3  |-  ( ph  ->  I  e. Ig )
41, 2, 3tethpnc 26070 . 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 3556 . 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 15 1  |-  ( ph  ->  P  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455   ` cfv 5255  PPointscpoints 26056  PLinescplines 26058  Igcig 26060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ig2 26061
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