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Theorem onnmin 4340
Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.)
Assertion
Ref Expression
onnmin  |-  ( ( A  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^| A )

Proof of Theorem onnmin
StepHypRef Expression
1 intss1 3672 . . 3  |-  ( B  e.  A  ->  |^| A  C_  B )
2 elirr 4313 . . . 4  |-  -.  B  e.  B
3 ssel 3003 . . . 4  |-  ( |^| A  C_  B  ->  ( B  e.  |^| A  ->  B  e.  B )
)
42, 3mtoi 623 . . 3  |-  ( |^| A  C_  B  ->  -.  B  e.  |^| A )
51, 4syl 14 . 2  |-  ( B  e.  A  ->  -.  B  e.  |^| A )
65adantl 271 1  |-  ( ( A  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^| A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    e. wcel 1434    C_ wss 2983   |^|cint 3657   Oncon0 4147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-setind 4309
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-v 2612  df-dif 2985  df-in 2989  df-ss 2996  df-sn 3423  df-int 3658
This theorem is referenced by: (None)
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