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Theorem el2oss1o 11356
Description: Being an element of ordinal two implies being a subset of ordinal one. The converse is equivalent to excluded middle by ss1oel2o 11357. (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 3454 . . 3  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
2 df2o3 6151 . . 3  |-  2o  =  { (/) ,  1o }
31, 2eleq2s 2179 . 2  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
4 0ss 3309 . . . 4  |-  (/)  C_  1o
5 sseq1 3036 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  1o  <->  (/)  C_  1o )
)
64, 5mpbiri 166 . . 3  |-  ( A  =  (/)  ->  A  C_  1o )
7 eqimss 3067 . . 3  |-  ( A  =  1o  ->  A  C_  1o )
86, 7jaoi 669 . 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 662    = wceq 1287    e. wcel 1436    C_ wss 2988   (/)c0 3275   {cpr 3432   1oc1o 6130   2oc2o 6131
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-sn 3437  df-pr 3438  df-suc 4174  df-1o 6137  df-2o 6138
This theorem is referenced by:  nninfsellemsuc  11373
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