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Theorem onint0 4603
Description: The intersection of a class of ordinal numbers is zero iff the class contains zero. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
onint0  |-  ( A 
C_  On  ->  ( |^| A  =  (/)  <->  (/)  e.  A
) )

Proof of Theorem onint0
StepHypRef Expression
1 0ex 4166 . . . . . . 7  |-  (/)  e.  _V
2 eleq1 2356 . . . . . . 7  |-  ( |^| A  =  (/)  ->  ( |^| A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 224 . . . . . 6  |-  ( |^| A  =  (/)  ->  |^| A  e.  _V )
4 intex 4183 . . . . . 6  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
53, 4sylibr 203 . . . . 5  |-  ( |^| A  =  (/)  ->  A  =/=  (/) )
6 onint 4602 . . . . 5  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
75, 6sylan2 460 . . . 4  |-  ( ( A  C_  On  /\  |^| A  =  (/) )  ->  |^| A  e.  A )
8 eleq1 2356 . . . . 5  |-  ( |^| A  =  (/)  ->  ( |^| A  e.  A  <->  (/)  e.  A
) )
98adantl 452 . . . 4  |-  ( ( A  C_  On  /\  |^| A  =  (/) )  -> 
( |^| A  e.  A  <->  (/)  e.  A ) )
107, 9mpbid 201 . . 3  |-  ( ( A  C_  On  /\  |^| A  =  (/) )  ->  (/) 
e.  A )
1110ex 423 . 2  |-  ( A 
C_  On  ->  ( |^| A  =  (/)  ->  (/)  e.  A
) )
12 int0el 3909 . 2  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
1311, 12impbid1 194 1  |-  ( A 
C_  On  ->  ( |^| A  =  (/)  <->  (/)  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    C_ wss 3165   (/)c0 3468   |^|cint 3878   Oncon0 4408
This theorem is referenced by:  cfeq0  7898
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-13 1698  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  ax-un 4528
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-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  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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