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Theorem snexxph 6806
Description: A case where the antecedent of snexg 4078 is not needed. The class  { x  | 
ph } is from dcextest 4465. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
Assertion
Ref Expression
snexxph  |-  { {
x  |  ph } }  e.  _V
Distinct variable group:    ph, x

Proof of Theorem snexxph
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 1on 6288 . . 3  |-  1o  e.  On
21elexi 2672 . 2  |-  1o  e.  _V
3 elsni 3515 . . . . 5  |-  ( y  e.  { { x  |  ph } }  ->  y  =  { x  | 
ph } )
4 vprc 4030 . . . . . . . 8  |-  -.  _V  e.  _V
5 df-v 2662 . . . . . . . . . 10  |-  _V  =  { x  |  x  =  x }
6 equid 1662 . . . . . . . . . . . 12  |-  x  =  x
7 pm5.1im 172 . . . . . . . . . . . 12  |-  ( x  =  x  ->  ( ph  ->  ( x  =  x  <->  ph ) ) )
86, 7ax-mp 5 . . . . . . . . . . 11  |-  ( ph  ->  ( x  =  x  <->  ph ) )
98abbidv 2235 . . . . . . . . . 10  |-  ( ph  ->  { x  |  x  =  x }  =  { x  |  ph }
)
105, 9syl5req 2163 . . . . . . . . 9  |-  ( ph  ->  { x  |  ph }  =  _V )
1110eleq1d 2186 . . . . . . . 8  |-  ( ph  ->  ( { x  | 
ph }  e.  _V  <->  _V  e.  _V ) )
124, 11mtbiri 649 . . . . . . 7  |-  ( ph  ->  -.  { x  | 
ph }  e.  _V )
13 19.8a 1554 . . . . . . . . 9  |-  ( y  =  { x  | 
ph }  ->  E. y 
y  =  { x  |  ph } )
143, 13syl 14 . . . . . . . 8  |-  ( y  e.  { { x  |  ph } }  ->  E. y  y  =  {
x  |  ph }
)
15 isset 2666 . . . . . . . 8  |-  ( { x  |  ph }  e.  _V  <->  E. y  y  =  { x  |  ph } )
1614, 15sylibr 133 . . . . . . 7  |-  ( y  e.  { { x  |  ph } }  ->  { x  |  ph }  e.  _V )
1712, 16nsyl3 600 . . . . . 6  |-  ( y  e.  { { x  |  ph } }  ->  -. 
ph )
18 vex 2663 . . . . . . . . . 10  |-  y  e. 
_V
19 biidd 171 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
2018, 19elab 2802 . . . . . . . . 9  |-  ( y  e.  { x  | 
ph }  <->  ph )
2120notbii 642 . . . . . . . 8  |-  ( -.  y  e.  { x  |  ph }  <->  -.  ph )
2221biimpri 132 . . . . . . 7  |-  ( -. 
ph  ->  -.  y  e.  { x  |  ph }
)
2322eq0rdv 3377 . . . . . 6  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
2417, 23syl 14 . . . . 5  |-  ( y  e.  { { x  |  ph } }  ->  { x  |  ph }  =  (/) )
253, 24eqtrd 2150 . . . 4  |-  ( y  e.  { { x  |  ph } }  ->  y  =  (/) )
26 0lt1o 6305 . . . 4  |-  (/)  e.  1o
2725, 26syl6eqel 2208 . . 3  |-  ( y  e.  { { x  |  ph } }  ->  y  e.  1o )
2827ssriv 3071 . 2  |-  { {
x  |  ph } }  C_  1o
292, 28ssexi 4036 1  |-  { {
x  |  ph } }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   _Vcvv 2660   (/)c0 3333   {csn 3497   Oncon0 4255   1oc1o 6274
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-1o 6281
This theorem is referenced by: (None)
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