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Theorem onnmin 4384
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 3703 . . 3  |-  ( B  e.  A  ->  |^| A  C_  B )
2 elirr 4357 . . . 4  |-  -.  B  e.  B
3 ssel 3019 . . . 4  |-  ( |^| A  C_  B  ->  ( B  e.  |^| A  ->  B  e.  B )
)
42, 3mtoi 625 . . 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 1438    C_ wss 2999   |^|cint 3688   Oncon0 4190
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-sn 3452  df-int 3689
This theorem is referenced by: (None)
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