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Theorem snexxph 6638
Description: A case where the antecedent of snexg 4010 is not needed. The class  { x  | 
ph } is from dcextest 4386. (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 6170 . . 3  |-  1o  e.  On
21elexi 2631 . 2  |-  1o  e.  _V
3 elsni 3459 . . . . 5  |-  ( y  e.  { { x  |  ph } }  ->  y  =  { x  | 
ph } )
4 vprc 3963 . . . . . . . 8  |-  -.  _V  e.  _V
5 df-v 2621 . . . . . . . . . 10  |-  _V  =  { x  |  x  =  x }
6 equid 1634 . . . . . . . . . . . 12  |-  x  =  x
7 pm5.1im 171 . . . . . . . . . . . 12  |-  ( x  =  x  ->  ( ph  ->  ( x  =  x  <->  ph ) ) )
86, 7ax-mp 7 . . . . . . . . . . 11  |-  ( ph  ->  ( x  =  x  <->  ph ) )
98abbidv 2205 . . . . . . . . . 10  |-  ( ph  ->  { x  |  x  =  x }  =  { x  |  ph }
)
105, 9syl5req 2133 . . . . . . . . 9  |-  ( ph  ->  { x  |  ph }  =  _V )
1110eleq1d 2156 . . . . . . . 8  |-  ( ph  ->  ( { x  | 
ph }  e.  _V  <->  _V  e.  _V ) )
124, 11mtbiri 635 . . . . . . 7  |-  ( ph  ->  -.  { x  | 
ph }  e.  _V )
13 19.8a 1527 . . . . . . . . 9  |-  ( y  =  { x  | 
ph }  ->  E. y 
y  =  { x  |  ph } )
143, 13syl 14 . . . . . . . 8  |-  ( y  e.  { { x  |  ph } }  ->  E. y  y  =  {
x  |  ph }
)
15 isset 2625 . . . . . . . 8  |-  ( { x  |  ph }  e.  _V  <->  E. y  y  =  { x  |  ph } )
1614, 15sylibr 132 . . . . . . 7  |-  ( y  e.  { { x  |  ph } }  ->  { x  |  ph }  e.  _V )
1712, 16nsyl3 591 . . . . . 6  |-  ( y  e.  { { x  |  ph } }  ->  -. 
ph )
18 vex 2622 . . . . . . . . . 10  |-  y  e. 
_V
19 biidd 170 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
2018, 19elab 2758 . . . . . . . . 9  |-  ( y  e.  { x  | 
ph }  <->  ph )
2120notbii 629 . . . . . . . 8  |-  ( -.  y  e.  { x  |  ph }  <->  -.  ph )
2221biimpri 131 . . . . . . 7  |-  ( -. 
ph  ->  -.  y  e.  { x  |  ph }
)
2322eq0rdv 3324 . . . . . 6  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
2417, 23syl 14 . . . . 5  |-  ( y  e.  { { x  |  ph } }  ->  { x  |  ph }  =  (/) )
253, 24eqtrd 2120 . . . 4  |-  ( y  e.  { { x  |  ph } }  ->  y  =  (/) )
26 0lt1o 6186 . . . 4  |-  (/)  e.  1o
2725, 26syl6eqel 2178 . . 3  |-  ( y  e.  { { x  |  ph } }  ->  y  e.  1o )
2827ssriv 3027 . 2  |-  { {
x  |  ph } }  C_  1o
292, 28ssexi 3969 1  |-  { {
x  |  ph } }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   _Vcvv 2619   (/)c0 3284   {csn 3441   Oncon0 4181   1oc1o 6156
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189  df-1o 6163
This theorem is referenced by: (None)
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