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Theorem onnmin 4594
Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)
Assertion
Ref Expression
onnmin  |-  ( ( A  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^| A )

Proof of Theorem onnmin
StepHypRef Expression
1 intss1 3877 . . 3  |-  ( B  e.  A  ->  |^| A  C_  B )
21adantl 452 . 2  |-  ( ( A  C_  On  /\  B  e.  A )  ->  |^| A  C_  B )
3 ne0i 3461 . . . 4  |-  ( B  e.  A  ->  A  =/=  (/) )
4 oninton 4591 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
53, 4sylan2 460 . . 3  |-  ( ( A  C_  On  /\  B  e.  A )  ->  |^| A  e.  On )
6 ssel2 3175 . . 3  |-  ( ( A  C_  On  /\  B  e.  A )  ->  B  e.  On )
7 ontri1 4426 . . 3  |-  ( (
|^| A  e.  On  /\  B  e.  On )  ->  ( |^| A  C_  B  <->  -.  B  e.  |^| A ) )
85, 6, 7syl2anc 642 . 2  |-  ( ( A  C_  On  /\  B  e.  A )  ->  ( |^| A  C_  B  <->  -.  B  e.  |^| A ) )
92, 8mpbid 201 1  |-  ( ( A  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^| A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   |^|cint 3862   Oncon0 4392
This theorem is referenced by:  onnminsb  4595  oneqmin  4596  onmindif2  4603  cardmin2  7631  ackbij1lem18  7863  cofsmo  7895  fin23lem26  7951  nofulllem5  24360
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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