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Theorem snon0 6725
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 4385 . . 3  |-  -.  A  e.  A
2 snidg 3493 . . . . . . 7  |-  ( A  e.  V  ->  A  e.  { A } )
32adantr 271 . . . . . 6  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  e.  { A } )
4 ontr1 4240 . . . . . . 7  |-  ( { A }  e.  On  ->  ( ( x  e.  A  /\  A  e. 
{ A } )  ->  x  e.  { A } ) )
54adantl 272 . . . . . 6  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( ( x  e.  A  /\  A  e.  { A } )  ->  x  e.  { A } ) )
63, 5mpan2d 420 . . . . 5  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( x  e.  A  ->  x  e.  { A } ) )
7 elsni 3484 . . . . 5  |-  ( x  e.  { A }  ->  x  =  A )
86, 7syl6 33 . . . 4  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  ( x  e.  A  ->  x  =  A ) )
9 eleq1 2157 . . . . 5  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
109biimpcd 158 . . . 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 628 . 2  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  -.  x  e.  A )
1312eq0rdv 3346 1  |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1296    e. wcel 1445   (/)c0 3302   {csn 3466   Oncon0 4214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219
This theorem is referenced by: (None)
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