MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oe0lem Unicode version

Theorem oe0lem 6512
Description: A helper lemma for oe0 6521 and others. (Contributed by NM, 6-Jan-2005.)
Hypotheses
Ref Expression
oe0lem.1  |-  ( (
ph  /\  A  =  (/) )  ->  ps )
oe0lem.2  |-  ( ( ( A  e.  On  /\ 
ph )  /\  (/)  e.  A
)  ->  ps )
Assertion
Ref Expression
oe0lem  |-  ( ( A  e.  On  /\  ph )  ->  ps )

Proof of Theorem oe0lem
StepHypRef Expression
1 oe0lem.1 . . . 4  |-  ( (
ph  /\  A  =  (/) )  ->  ps )
21ex 423 . . 3  |-  ( ph  ->  ( A  =  (/)  ->  ps ) )
32adantl 452 . 2  |-  ( ( A  e.  On  /\  ph )  ->  ( A  =  (/)  ->  ps )
)
4 on0eln0 4447 . . . 4  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
54adantr 451 . . 3  |-  ( ( A  e.  On  /\  ph )  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
6 oe0lem.2 . . . 4  |-  ( ( ( A  e.  On  /\ 
ph )  /\  (/)  e.  A
)  ->  ps )
76ex 423 . . 3  |-  ( ( A  e.  On  /\  ph )  ->  ( (/)  e.  A  ->  ps ) )
85, 7sylbird 226 . 2  |-  ( ( A  e.  On  /\  ph )  ->  ( A  =/=  (/)  ->  ps )
)
93, 8pm2.61dne 2523 1  |-  ( ( A  e.  On  /\  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   Oncon0 4392
This theorem is referenced by:  oe0  6521  oev2  6522  oesuclem  6524  oecl  6536  odi  6577  oewordri  6590  oelim2  6593  oeoa  6595  oeoe  6597
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
  Copyright terms: Public domain W3C validator