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Theorem onnmin 4775
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 4057 . . 3  |-  ( B  e.  A  ->  |^| A  C_  B )
21adantl 453 . 2  |-  ( ( A  C_  On  /\  B  e.  A )  ->  |^| A  C_  B )
3 ne0i 3626 . . . 4  |-  ( B  e.  A  ->  A  =/=  (/) )
4 oninton 4772 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
53, 4sylan2 461 . . 3  |-  ( ( A  C_  On  /\  B  e.  A )  ->  |^| A  e.  On )
6 ssel2 3335 . . 3  |-  ( ( A  C_  On  /\  B  e.  A )  ->  B  e.  On )
7 ontri1 4607 . . 3  |-  ( (
|^| A  e.  On  /\  B  e.  On )  ->  ( |^| A  C_  B  <->  -.  B  e.  |^| A ) )
85, 6, 7syl2anc 643 . 2  |-  ( ( A  C_  On  /\  B  e.  A )  ->  ( |^| A  C_  B  <->  -.  B  e.  |^| A ) )
92, 8mpbid 202 1  |-  ( ( A  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^| A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   |^|cint 4042   Oncon0 4573
This theorem is referenced by:  onnminsb  4776  oneqmin  4777  onmindif2  4784  cardmin2  7875  ackbij1lem18  8107  cofsmo  8139  fin23lem26  8195  nofulllem5  25626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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