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Theorem onnmin 6527
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 4254 . . 3  |-  ( B  e.  A  ->  |^| A  C_  B )
21adantl 466 . 2  |-  ( ( A  C_  On  /\  B  e.  A )  ->  |^| A  C_  B )
3 ne0i 3754 . . . 4  |-  ( B  e.  A  ->  A  =/=  (/) )
4 oninton 6524 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
53, 4sylan2 474 . . 3  |-  ( ( A  C_  On  /\  B  e.  A )  ->  |^| A  e.  On )
6 ssel2 3462 . . 3  |-  ( ( A  C_  On  /\  B  e.  A )  ->  B  e.  On )
7 ontri1 4864 . . 3  |-  ( (
|^| A  e.  On  /\  B  e.  On )  ->  ( |^| A  C_  B  <->  -.  B  e.  |^| A ) )
85, 6, 7syl2anc 661 . 2  |-  ( ( A  C_  On  /\  B  e.  A )  ->  ( |^| A  C_  B  <->  -.  B  e.  |^| A ) )
92, 8mpbid 210 1  |-  ( ( A  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^| A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758    =/= wne 2648    C_ wss 3439   (/)c0 3748   |^|cint 4239   Oncon0 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834
This theorem is referenced by:  onnminsb  6528  oneqmin  6529  onmindif2  6536  cardmin2  8282  ackbij1lem18  8520  cofsmo  8552  fin23lem26  8608  nofulllem5  28011
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