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Theorem snon0 6573
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 4323 . . 3  |-  -.  A  e.  A
2 snidg 3450 . . . . . . 7  |-  ( A  e.  V  ->  A  e.  { A } )
32adantr 270 . . . . . 6  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  e.  { A } )
4 ontr1 4183 . . . . . . 7  |-  ( { A }  e.  On  ->  ( ( x  e.  A  /\  A  e. 
{ A } )  ->  x  e.  { A } ) )
54adantl 271 . . . . . 6  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( ( x  e.  A  /\  A  e.  { A } )  ->  x  e.  { A } ) )
63, 5mpan2d 419 . . . . 5  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( x  e.  A  ->  x  e.  { A } ) )
7 elsni 3443 . . . . 5  |-  ( x  e.  { A }  ->  x  =  A )
86, 7syl6 33 . . . 4  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( x  e.  A  ->  x  =  A ) )
9 eleq1 2147 . . . . 5  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
109biimpcd 157 . . . 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 623 . 2  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  -.  x  e.  A )
1312eq0rdv 3312 1  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436   (/)c0 3272   {csn 3425   Oncon0 4157
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  ax-setind 4319
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2616  df-dif 2988  df-in 2992  df-ss 2999  df-nul 3273  df-sn 3431  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162
This theorem is referenced by: (None)
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