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Theorem snon0 6966
Description: An ordinal which is a singleton is  { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
Assertion
Ref Expression
snon0  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  =  (/) )

Proof of Theorem snon0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elirr 4558 . . 3  |-  -.  A  e.  A
2 snidg 3636 . . . . . . 7  |-  ( A  e.  V  ->  A  e.  { A } )
32adantr 276 . . . . . 6  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  e.  { A } )
4 ontr1 4407 . . . . . . 7  |-  ( { A }  e.  On  ->  ( ( x  e.  A  /\  A  e. 
{ A } )  ->  x  e.  { A } ) )
54adantl 277 . . . . . 6  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( ( x  e.  A  /\  A  e.  { A } )  ->  x  e.  { A } ) )
63, 5mpan2d 428 . . . . 5  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( x  e.  A  ->  x  e.  { A } ) )
7 elsni 3625 . . . . 5  |-  ( x  e.  { A }  ->  x  =  A )
86, 7syl6 33 . . . 4  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( x  e.  A  ->  x  =  A ) )
9 eleq1 2252 . . . . 5  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
109biimpcd 159 . . . 4  |-  ( x  e.  A  ->  (
x  =  A  ->  A  e.  A )
)
118, 10sylcom 28 . . 3  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( x  e.  A  ->  A  e.  A ) )
121, 11mtoi 665 . 2  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  -.  x  e.  A )
1312eq0rdv 3482 1  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   (/)c0 3437   {csn 3607   Oncon0 4381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386
This theorem is referenced by: (None)
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