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Theorem oe0lem 6759
Description: A helper lemma for oe0 6768 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 425 . . 3  |-  ( ph  ->  ( A  =  (/)  ->  ps ) )
32adantl 454 . 2  |-  ( ( A  e.  On  /\  ph )  ->  ( A  =  (/)  ->  ps )
)
4 on0eln0 4638 . . . 4  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
54adantr 453 . . 3  |-  ( ( A  e.  On  /\  ph )  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
6 oe0lem.2 . . . 4  |-  ( ( ( A  e.  On  /\ 
ph )  /\  (/)  e.  A
)  ->  ps )
76ex 425 . . 3  |-  ( ( A  e.  On  /\  ph )  ->  ( (/)  e.  A  ->  ps ) )
85, 7sylbird 228 . 2  |-  ( ( A  e.  On  /\  ph )  ->  ( A  =/=  (/)  ->  ps )
)
93, 8pm2.61dne 2683 1  |-  ( ( A  e.  On  /\  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   Oncon0 4583
This theorem is referenced by:  oe0  6768  oev2  6769  oesuclem  6771  oecl  6783  odi  6824  oewordri  6837  oelim2  6840  oeoa  6842  oeoe  6844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
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