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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

Proof of Theorem el2oss1o
StepHypRef Expression
1 elpri 3518 . . 3
2 df2o3 6293 . . 3
31, 2eleq2s 2210 . 2
4 0ss 3369 . . . 4
5 sseq1 3088 . . . 4
64, 5mpbiri 167 . . 3
7 eqimss 3119 . . 3
86, 7jaoi 688 . 2
93, 8syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 680   wceq 1314   wcel 1463   wss 3039  c0 3331  cpr 3496  c1o 6272  c2o 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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