ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snexxph Unicode version

Theorem snexxph 6804
Description: A case where the antecedent of snexg 4076 is not needed. The class  { x  | 
ph } is from dcextest 4463. (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 6286 . . 3  |-  1o  e.  On
21elexi 2670 . 2  |-  1o  e.  _V
3 elsni 3513 . . . . 5  |-  ( y  e.  { { x  |  ph } }  ->  y  =  { x  | 
ph } )
4 vprc 4028 . . . . . . . 8  |-  -.  _V  e.  _V
5 df-v 2660 . . . . . . . . . 10  |-  _V  =  { x  |  x  =  x }
6 equid 1660 . . . . . . . . . . . 12  |-  x  =  x
7 pm5.1im 172 . . . . . . . . . . . 12  |-  ( x  =  x  ->  ( ph  ->  ( x  =  x  <->  ph ) ) )
86, 7ax-mp 5 . . . . . . . . . . 11  |-  ( ph  ->  ( x  =  x  <->  ph ) )
98abbidv 2233 . . . . . . . . . 10  |-  ( ph  ->  { x  |  x  =  x }  =  { x  |  ph }
)
105, 9syl5req 2161 . . . . . . . . 9  |-  ( ph  ->  { x  |  ph }  =  _V )
1110eleq1d 2184 . . . . . . . 8  |-  ( ph  ->  ( { x  | 
ph }  e.  _V  <->  _V  e.  _V ) )
124, 11mtbiri 647 . . . . . . 7  |-  ( ph  ->  -.  { x  | 
ph }  e.  _V )
13 19.8a 1552 . . . . . . . . 9  |-  ( y  =  { x  | 
ph }  ->  E. y 
y  =  { x  |  ph } )
143, 13syl 14 . . . . . . . 8  |-  ( y  e.  { { x  |  ph } }  ->  E. y  y  =  {
x  |  ph }
)
15 isset 2664 . . . . . . . 8  |-  ( { x  |  ph }  e.  _V  <->  E. y  y  =  { x  |  ph } )
1614, 15sylibr 133 . . . . . . 7  |-  ( y  e.  { { x  |  ph } }  ->  { x  |  ph }  e.  _V )
1712, 16nsyl3 598 . . . . . 6  |-  ( y  e.  { { x  |  ph } }  ->  -. 
ph )
18 vex 2661 . . . . . . . . . 10  |-  y  e. 
_V
19 biidd 171 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ph 
<-> 
ph ) )
2018, 19elab 2800 . . . . . . . . 9  |-  ( y  e.  { x  | 
ph }  <->  ph )
2120notbii 640 . . . . . . . 8  |-  ( -.  y  e.  { x  |  ph }  <->  -.  ph )
2221biimpri 132 . . . . . . 7  |-  ( -. 
ph  ->  -.  y  e.  { x  |  ph }
)
2322eq0rdv 3375 . . . . . 6  |-  ( -. 
ph  ->  { x  | 
ph }  =  (/) )
2417, 23syl 14 . . . . 5  |-  ( y  e.  { { x  |  ph } }  ->  { x  |  ph }  =  (/) )
253, 24eqtrd 2148 . . . 4  |-  ( y  e.  { { x  |  ph } }  ->  y  =  (/) )
26 0lt1o 6303 . . . 4  |-  (/)  e.  1o
2725, 26syl6eqel 2206 . . 3  |-  ( y  e.  { { x  |  ph } }  ->  y  e.  1o )
2827ssriv 3069 . 2  |-  { {
x  |  ph } }  C_  1o
292, 28ssexi 4034 1  |-  { {
x  |  ph } }  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   _Vcvv 2658   (/)c0 3331   {csn 3495   Oncon0 4253   1oc1o 6272
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-tr 3995  df-iord 4256  df-on 4258  df-suc 4261  df-1o 6279
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator