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Theorem onint0 4587
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 4150 . . . . . . 7  |-  (/)  e.  _V
2 eleq1 2343 . . . . . . 7  |-  ( |^| A  =  (/)  ->  ( |^| A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 224 . . . . . 6  |-  ( |^| A  =  (/)  ->  |^| A  e.  _V )
4 intex 4167 . . . . . 6  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
53, 4sylibr 203 . . . . 5  |-  ( |^| A  =  (/)  ->  A  =/=  (/) )
6 onint 4586 . . . . 5  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
75, 6sylan2 460 . . . 4  |-  ( ( A  C_  On  /\  |^| A  =  (/) )  ->  |^| A  e.  A )
8 eleq1 2343 . . . . 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 3893 . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   |^|cint 3862   Oncon0 4392
This theorem is referenced by:  cfeq0  7882
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-13 1686  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  ax-un 4512
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-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  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
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