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Theorem onnmin 4443
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 3752 . . 3  |-  ( B  e.  A  ->  |^| A  C_  B )
2 elirr 4416 . . . 4  |-  -.  B  e.  B
3 ssel 3057 . . . 4  |-  ( |^| A  C_  B  ->  ( B  e.  |^| A  ->  B  e.  B )
)
42, 3mtoi 636 . . 3  |-  ( |^| A  C_  B  ->  -.  B  e.  |^| A )
51, 4syl 14 . 2  |-  ( B  e.  A  ->  -.  B  e.  |^| A )
65adantl 273 1  |-  ( ( A  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^| A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 1463    C_ wss 3037   |^|cint 3737   Oncon0 4245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-setind 4412
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-v 2659  df-dif 3039  df-in 3043  df-ss 3050  df-sn 3499  df-int 3738
This theorem is referenced by: (None)
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