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Theorem snexxph 6943
Description: A case where the antecedent of snexg 4181 is not needed. The class  { x  | 
ph } is from dcextest 4577. (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 6418 . . 3  |-  1o  e.  On
21elexi 2749 . 2  |-  1o  e.  _V
3 elsni 3609 . . . . 5  |-  ( y  e.  { { x  |  ph } }  ->  y  =  { x  | 
ph } )
4 vprc 4132 . . . . . . . 8  |-  -.  _V  e.  _V
5 df-v 2739 . . . . . . . . . 10  |-  _V  =  { x  |  x  =  x }
6 equid 1701 . . . . . . . . . . . 12  |-  x  =  x
7 pm5.1im 173 . . . . . . . . . . . 12  |-  ( x  =  x  ->  ( ph  ->  ( x  =  x  <->  ph ) ) )
86, 7ax-mp 5 . . . . . . . . . . 11  |-  ( ph  ->  ( x  =  x  <->  ph ) )
98abbidv 2295 . . . . . . . . . 10  |-  ( ph  ->  { x  |  x  =  x }  =  { x  |  ph }
)
105, 9eqtr2id 2223 . . . . . . . . 9  |-  ( ph  ->  { x  |  ph }  =  _V )
1110eleq1d 2246 . . . . . . . 8  |-  ( ph  ->  ( { x  | 
ph }  e.  _V  <->  _V  e.  _V ) )
124, 11mtbiri 675 . . . . . . 7  |-  ( ph  ->  -.  { x  | 
ph }  e.  _V )
13 19.8a 1590 . . . . . . . . 9  |-  ( y  =  { x  | 
ph }  ->  E. y 
y  =  { x  |  ph } )
143, 13syl 14 . . . . . . . 8  |-  ( y  e.  { { x  |  ph } }  ->  E. y  y  =  {
x  |  ph }
)
15 isset 2743 . . . . . . . 8  |-  ( { x  |  ph }  e.  _V  <->  E. y  y  =  { x  |  ph } )
1614, 15sylibr 134 . . . . . . 7  |-  ( y  e.  { { x  |  ph } }  ->  { x  |  ph }  e.  _V )
1712, 16nsyl3 626 . . . . . 6  |-  ( y  e.  { { x  |  ph } }  ->  -. 
ph )
18 vex 2740 . . . . . . . . . 10  |-  y  e. 
_V
19 biidd 172 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
2018, 19elab 2881 . . . . . . . . 9  |-  ( y  e.  { x  | 
ph }  <->  ph )
2120notbii 668 . . . . . . . 8  |-  ( -.  y  e.  { x  |  ph }  <->  -.  ph )
2221biimpri 133 . . . . . . 7  |-  ( -. 
ph  ->  -.  y  e.  { x  |  ph }
)
2322eq0rdv 3467 . . . . . 6  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
2417, 23syl 14 . . . . 5  |-  ( y  e.  { { x  |  ph } }  ->  { x  |  ph }  =  (/) )
253, 24eqtrd 2210 . . . 4  |-  ( y  e.  { { x  |  ph } }  ->  y  =  (/) )
26 0lt1o 6435 . . . 4  |-  (/)  e.  1o
2725, 26eqeltrdi 2268 . . 3  |-  ( y  e.  { { x  |  ph } }  ->  y  e.  1o )
2827ssriv 3159 . 2  |-  { {
x  |  ph } }  C_  1o
292, 28ssexi 4138 1  |-  { {
x  |  ph } }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   _Vcvv 2737   (/)c0 3422   {csn 3591   Oncon0 4360   1oc1o 6404
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-tr 4099  df-iord 4363  df-on 4365  df-suc 4368  df-1o 6411
This theorem is referenced by: (None)
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