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Theorem ss1oel2o 11888
Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4032 which more directly illustrates the contrast with el2oss1o 11887. (Contributed by Jim Kingdon, 8-Aug-2022.)
Assertion
Ref Expression
ss1oel2o  |-  (EXMID  <->  A. x
( x  C_  1o  ->  x  e.  2o ) )

Proof of Theorem ss1oel2o
StepHypRef Expression
1 exmid01 4032 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
2 df1o2 6194 . . . . 5  |-  1o  =  { (/) }
32sseq2i 3051 . . . 4  |-  ( x 
C_  1o  <->  x  C_  { (/) } )
4 df2o2 6196 . . . . . 6  |-  2o  =  { (/) ,  { (/) } }
54eleq2i 2154 . . . . 5  |-  ( x  e.  2o  <->  x  e.  {
(/) ,  { (/) } }
)
6 vex 2622 . . . . . 6  |-  x  e. 
_V
76elpr 3467 . . . . 5  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
85, 7bitri 182 . . . 4  |-  ( x  e.  2o  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
93, 8imbi12i 237 . . 3  |-  ( ( x  C_  1o  ->  x  e.  2o )  <->  ( x  C_ 
{ (/) }  ->  (
x  =  (/)  \/  x  =  { (/) } ) ) )
109albii 1404 . 2  |-  ( A. x ( x  C_  1o  ->  x  e.  2o ) 
<-> 
A. x ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
111, 10bitr4i 185 1  |-  (EXMID  <->  A. x
( x  C_  1o  ->  x  e.  2o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 664   A.wal 1287    = wceq 1289    e. wcel 1438    C_ wss 2999   (/)c0 3286   {csn 3446   {cpr 3447  EXMIDwem 4029   1oc1o 6174   2oc2o 6175
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-nul 3965
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-pr 3453  df-exmid 4030  df-suc 4198  df-1o 6181  df-2o 6182
This theorem is referenced by: (None)
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