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Theorem onint0 4777
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 4340 . . . . . . 7  |-  (/)  e.  _V
2 eleq1 2497 . . . . . . 7  |-  ( |^| A  =  (/)  ->  ( |^| A  e.  _V  <->  (/)  e.  _V ) )
31, 2mpbiri 226 . . . . . 6  |-  ( |^| A  =  (/)  ->  |^| A  e.  _V )
4 intex 4357 . . . . . 6  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
53, 4sylibr 205 . . . . 5  |-  ( |^| A  =  (/)  ->  A  =/=  (/) )
6 onint 4776 . . . . 5  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
75, 6sylan2 462 . . . 4  |-  ( ( A  C_  On  /\  |^| A  =  (/) )  ->  |^| A  e.  A )
8 eleq1 2497 . . . . 5  |-  ( |^| A  =  (/)  ->  ( |^| A  e.  A  <->  (/)  e.  A
) )
98adantl 454 . . . 4  |-  ( ( A  C_  On  /\  |^| A  =  (/) )  -> 
( |^| A  e.  A  <->  (/)  e.  A ) )
107, 9mpbid 203 . . 3  |-  ( ( A  C_  On  /\  |^| A  =  (/) )  ->  (/) 
e.  A )
1110ex 425 . 2  |-  ( A 
C_  On  ->  ( |^| A  =  (/)  ->  (/)  e.  A
) )
12 int0el 4082 . 2  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
1311, 12impbid1 196 1  |-  ( A 
C_  On  ->  ( |^| A  =  (/)  <->  (/)  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   _Vcvv 2957    C_ wss 3321   (/)c0 3629   |^|cint 4051   Oncon0 4582
This theorem is referenced by:  cfeq0  8137
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-br 4214  df-opab 4268  df-tr 4304  df-eprel 4495  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586
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