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Theorem el2oss1o 12999
Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 13000. (Contributed by Jim Kingdon, 8-Aug-2022.)
Assertion
Ref Expression
el2oss1o  |-  ( A  e.  2o  ->  A  C_  1o )

Proof of Theorem el2oss1o
StepHypRef Expression
1 elpri 3518 . . 3  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
2 df2o3 6293 . . 3  |-  2o  =  { (/) ,  1o }
31, 2eleq2s 2210 . 2  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
4 0ss 3369 . . . 4  |-  (/)  C_  1o
5 sseq1 3088 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  1o  <->  (/)  C_  1o )
)
64, 5mpbiri 167 . . 3  |-  ( A  =  (/)  ->  A  C_  1o )
7 eqimss 3119 . . 3  |-  ( A  =  1o  ->  A  C_  1o )
86, 7jaoi 688 . 2  |-  ( ( A  =  (/)  \/  A  =  1o )  ->  A  C_  1o )
93, 8syl 14 1  |-  ( A  e.  2o  ->  A  C_  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 680    = wceq 1314    e. wcel 1463    C_ wss 3039   (/)c0 3331   {cpr 3496   1oc1o 6272   2oc2o 6273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 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
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-suc 4261  df-1o 6279  df-2o 6280
This theorem is referenced by:  nninfsellemsuc  13019
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