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Theorem onnmin 6574
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 4239 . . 3  |-  ( B  e.  A  ->  |^| A  C_  B )
21adantl 464 . 2  |-  ( ( A  C_  On  /\  B  e.  A )  ->  |^| A  C_  B )
3 ne0i 3741 . . . 4  |-  ( B  e.  A  ->  A  =/=  (/) )
4 oninton 6571 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
53, 4sylan2 472 . . 3  |-  ( ( A  C_  On  /\  B  e.  A )  ->  |^| A  e.  On )
6 ssel2 3434 . . 3  |-  ( ( A  C_  On  /\  B  e.  A )  ->  B  e.  On )
7 ontri1 4853 . . 3  |-  ( (
|^| A  e.  On  /\  B  e.  On )  ->  ( |^| A  C_  B  <->  -.  B  e.  |^| A ) )
85, 6, 7syl2anc 659 . 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 367    e. wcel 1840    =/= wne 2596    C_ wss 3411   (/)c0 3735   |^|cint 4224   Oncon0 4819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-br 4393  df-opab 4451  df-tr 4487  df-eprel 4731  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823
This theorem is referenced by:  onnminsb  6575  oneqmin  6576  onmindif2  6583  cardmin2  8329  ackbij1lem18  8567  cofsmo  8599  fin23lem26  8655  nofulllem5  30134
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