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Theorem oe0lem 6716
Description: A helper lemma for oe0 6725 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 424 . . 3  |-  ( ph  ->  ( A  =  (/)  ->  ps ) )
32adantl 453 . 2  |-  ( ( A  e.  On  /\  ph )  ->  ( A  =  (/)  ->  ps )
)
4 on0eln0 4596 . . . 4  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
54adantr 452 . . 3  |-  ( ( A  e.  On  /\  ph )  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
6 oe0lem.2 . . . 4  |-  ( ( ( A  e.  On  /\ 
ph )  /\  (/)  e.  A
)  ->  ps )
76ex 424 . . 3  |-  ( ( A  e.  On  /\  ph )  ->  ( (/)  e.  A  ->  ps ) )
85, 7sylbird 227 . 2  |-  ( ( A  e.  On  /\  ph )  ->  ( A  =/=  (/)  ->  ps )
)
93, 8pm2.61dne 2644 1  |-  ( ( A  e.  On  /\  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   (/)c0 3588   Oncon0 4541
This theorem is referenced by:  oe0  6725  oev2  6726  oesuclem  6728  oecl  6740  odi  6781  oewordri  6794  oelim2  6797  oeoa  6799  oeoe  6801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545
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