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Theorem exss 4252
Description: Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
Assertion
Ref Expression
exss  |-  ( E. x  e.  A  ph  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
Distinct variable groups:    x, y, A    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem exss
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-rab 2565 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21neeq1i 2469 . . 3  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
3 rabn0 3487 . . 3  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
4 n0 3477 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  =/=  (/)  <->  E. z 
z  e.  { x  |  ( x  e.  A  /\  ph ) } )
52, 3, 43bitr3i 266 . 2  |-  ( E. x  e.  A  ph  <->  E. z  z  e.  {
x  |  ( x  e.  A  /\  ph ) } )
6 vex 2804 . . . . . 6  |-  z  e. 
_V
76snss 3761 . . . . 5  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  { z }  C_  { x  |  ( x  e.  A  /\  ph ) } )
8 ssab2 3270 . . . . . 6  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
9 sstr2 3199 . . . . . 6  |-  ( { z }  C_  { x  |  ( x  e.  A  /\  ph ) }  ->  ( { x  |  ( x  e.  A  /\  ph ) }  C_  A  ->  { z }  C_  A )
)
108, 9mpi 16 . . . . 5  |-  ( { z }  C_  { x  |  ( x  e.  A  /\  ph ) }  ->  { z } 
C_  A )
117, 10sylbi 187 . . . 4  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  { z }  C_  A
)
12 simpr 447 . . . . . . . 8  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  ->  [ z  /  x ] ph )
13 equsb1 1987 . . . . . . . . 9  |-  [ z  /  x ] x  =  z
14 elsn 3668 . . . . . . . . . 10  |-  ( x  e.  { z }  <-> 
x  =  z )
1514sbbii 1643 . . . . . . . . 9  |-  ( [ z  /  x ]
x  e.  { z }  <->  [ z  /  x ] x  =  z
)
1613, 15mpbir 200 . . . . . . . 8  |-  [ z  /  x ] x  e.  { z }
1712, 16jctil 523 . . . . . . 7  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  ->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
18 df-clab 2283 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  [ z  /  x ] ( x  e.  A  /\  ph ) )
19 sban 2022 . . . . . . . 8  |-  ( [ z  /  x ]
( x  e.  A  /\  ph )  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
2018, 19bitri 240 . . . . . . 7  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
21 df-rab 2565 . . . . . . . . 9  |-  { x  e.  { z }  |  ph }  =  { x  |  ( x  e. 
{ z }  /\  ph ) }
2221eleq2i 2360 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  z  e.  {
x  |  ( x  e.  { z }  /\  ph ) } )
23 df-clab 2283 . . . . . . . . 9  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  [ z  /  x ] ( x  e. 
{ z }  /\  ph ) )
24 sban 2022 . . . . . . . . 9  |-  ( [ z  /  x ]
( x  e.  {
z }  /\  ph ) 
<->  ( [ z  /  x ] x  e.  {
z }  /\  [
z  /  x ] ph ) )
2523, 24bitri 240 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
2622, 25bitri 240 . . . . . . 7  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
2717, 20, 263imtr4i 257 . . . . . 6  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  z  e.  { x  e. 
{ z }  |  ph } )
28 ne0i 3474 . . . . . 6  |-  ( z  e.  { x  e. 
{ z }  |  ph }  ->  { x  e.  { z }  |  ph }  =/=  (/) )
2927, 28syl 15 . . . . 5  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  { x  e.  { z }  |  ph }  =/=  (/) )
30 rabn0 3487 . . . . 5  |-  ( { x  e.  { z }  |  ph }  =/=  (/)  <->  E. x  e.  {
z } ph )
3129, 30sylib 188 . . . 4  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. x  e.  { z } ph )
32 snex 4232 . . . . 5  |-  { z }  e.  _V
33 sseq1 3212 . . . . . 6  |-  ( y  =  { z }  ->  ( y  C_  A 
<->  { z }  C_  A ) )
34 rexeq 2750 . . . . . 6  |-  ( y  =  { z }  ->  ( E. x  e.  y  ph  <->  E. x  e.  { z } ph ) )
3533, 34anbi12d 691 . . . . 5  |-  ( y  =  { z }  ->  ( ( y 
C_  A  /\  E. x  e.  y  ph ) 
<->  ( { z } 
C_  A  /\  E. x  e.  { z } ph ) ) )
3632, 35spcev 2888 . . . 4  |-  ( ( { z }  C_  A  /\  E. x  e. 
{ z } ph )  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
3711, 31, 36syl2anc 642 . . 3  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. y ( y  C_  A  /\  E. x  e.  y  ph ) )
3837exlimiv 1624 . 2  |-  ( E. z  z  e.  {
x  |  ( x  e.  A  /\  ph ) }  ->  E. y
( y  C_  A  /\  E. x  e.  y 
ph ) )
395, 38sylbi 187 1  |-  ( E. x  e.  A  ph  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660
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