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Theorem oe0lem 6528
Description: A helper lemma for oe0 6537 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 4463 . . . 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 2536 1  |-  ( ( A  e.  On  /\  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   Oncon0 4408
This theorem is referenced by:  oe0  6537  oev2  6538  oesuclem  6540  oecl  6552  odi  6593  oewordri  6606  oelim2  6609  oeoa  6611  oeoe  6613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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