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Theorem ss1oel2o 13023
Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4089 which more directly illustrates the contrast with el2oss1o 13022. (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 4089 . 2  |-  (EXMID  <->  A. x
( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/)
} ) ) )
2 df1o2 6292 . . . . 5  |-  1o  =  { (/) }
32sseq2i 3092 . . . 4  |-  ( x 
C_  1o  <->  x  C_  { (/) } )
4 df2o2 6294 . . . . . 6  |-  2o  =  { (/) ,  { (/) } }
54eleq2i 2182 . . . . 5  |-  ( x  e.  2o  <->  x  e.  {
(/) ,  { (/) } }
)
6 vex 2661 . . . . . 6  |-  x  e. 
_V
76elpr 3516 . . . . 5  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
85, 7bitri 183 . . . 4  |-  ( x  e.  2o  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
93, 8imbi12i 238 . . 3  |-  ( ( x  C_  1o  ->  x  e.  2o )  <->  ( x  C_ 
{ (/) }  ->  (
x  =  (/)  \/  x  =  { (/) } ) ) )
109albii 1429 . 2  |-  ( A. x ( x  C_  1o  ->  x  e.  2o ) 
<-> 
A. x ( x 
C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
111, 10bitr4i 186 1  |-  (EXMID  <->  A. x
( x  C_  1o  ->  x  e.  2o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 680   A.wal 1312    = wceq 1314    e. wcel 1463    C_ wss 3039   (/)c0 3331   {csn 3495   {cpr 3496  EXMIDwem 4086   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  ax-nul 4022
This theorem depends on definitions:  df-bi 116  df-dc 803  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-exmid 4087  df-suc 4261  df-1o 6279  df-2o 6280
This theorem is referenced by: (None)
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