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Theorem exss 4656
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 2804 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21neeq1i 2733 . . 3  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
3 rabn0 3758 . . 3  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
4 n0 3747 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  =/=  (/)  <->  E. z 
z  e.  { x  |  ( x  e.  A  /\  ph ) } )
52, 3, 43bitr3i 275 . 2  |-  ( E. x  e.  A  ph  <->  E. z  z  e.  {
x  |  ( x  e.  A  /\  ph ) } )
6 vex 3074 . . . . . 6  |-  z  e. 
_V
76snss 4100 . . . . 5  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  { z }  C_  { x  |  ( x  e.  A  /\  ph ) } )
8 ssab2 3537 . . . . . 6  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
9 sstr2 3464 . . . . . 6  |-  ( { z }  C_  { x  |  ( x  e.  A  /\  ph ) }  ->  ( { x  |  ( x  e.  A  /\  ph ) }  C_  A  ->  { z }  C_  A )
)
108, 9mpi 17 . . . . 5  |-  ( { z }  C_  { x  |  ( x  e.  A  /\  ph ) }  ->  { z } 
C_  A )
117, 10sylbi 195 . . . 4  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  { z }  C_  A
)
12 simpr 461 . . . . . . . 8  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  ->  [ z  /  x ] ph )
13 equsb1 2064 . . . . . . . . 9  |-  [ z  /  x ] x  =  z
14 elsn 3992 . . . . . . . . . 10  |-  ( x  e.  { z }  <-> 
x  =  z )
1514sbbii 1709 . . . . . . . . 9  |-  ( [ z  /  x ]
x  e.  { z }  <->  [ z  /  x ] x  =  z
)
1613, 15mpbir 209 . . . . . . . 8  |-  [ z  /  x ] x  e.  { z }
1712, 16jctil 537 . . . . . . 7  |-  ( ( [ z  /  x ] x  e.  A  /\  [ z  /  x ] ph )  ->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
18 df-clab 2437 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  [ z  /  x ] ( x  e.  A  /\  ph ) )
19 sban 2098 . . . . . . . 8  |-  ( [ z  /  x ]
( x  e.  A  /\  ph )  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
2018, 19bitri 249 . . . . . . 7  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  ( [
z  /  x ]
x  e.  A  /\  [ z  /  x ] ph ) )
21 df-rab 2804 . . . . . . . . 9  |-  { x  e.  { z }  |  ph }  =  { x  |  ( x  e. 
{ z }  /\  ph ) }
2221eleq2i 2529 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  z  e.  {
x  |  ( x  e.  { z }  /\  ph ) } )
23 df-clab 2437 . . . . . . . . 9  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  [ z  /  x ] ( x  e. 
{ z }  /\  ph ) )
24 sban 2098 . . . . . . . . 9  |-  ( [ z  /  x ]
( x  e.  {
z }  /\  ph ) 
<->  ( [ z  /  x ] x  e.  {
z }  /\  [
z  /  x ] ph ) )
2523, 24bitri 249 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
2622, 25bitri 249 . . . . . . 7  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  ( [ z  /  x ] x  e.  { z }  /\  [ z  /  x ] ph ) )
2717, 20, 263imtr4i 266 . . . . . 6  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  z  e.  { x  e. 
{ z }  |  ph } )
28 ne0i 3744 . . . . . 6  |-  ( z  e.  { x  e. 
{ z }  |  ph }  ->  { x  e.  { z }  |  ph }  =/=  (/) )
2927, 28syl 16 . . . . 5  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  { x  e.  { z }  |  ph }  =/=  (/) )
30 rabn0 3758 . . . . 5  |-  ( { x  e.  { z }  |  ph }  =/=  (/)  <->  E. x  e.  {
z } ph )
3129, 30sylib 196 . . . 4  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. x  e.  { z } ph )
32 snex 4634 . . . . 5  |-  { z }  e.  _V
33 sseq1 3478 . . . . . 6  |-  ( y  =  { z }  ->  ( y  C_  A 
<->  { z }  C_  A ) )
34 rexeq 3017 . . . . . 6  |-  ( y  =  { z }  ->  ( E. x  e.  y  ph  <->  E. x  e.  { z } ph ) )
3533, 34anbi12d 710 . . . . 5  |-  ( y  =  { z }  ->  ( ( y 
C_  A  /\  E. x  e.  y  ph ) 
<->  ( { z } 
C_  A  /\  E. x  e.  { z } ph ) ) )
3632, 35spcev 3163 . . . 4  |-  ( ( { z }  C_  A  /\  E. x  e. 
{ z } ph )  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
3711, 31, 36syl2anc 661 . . 3  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  ->  E. y ( y  C_  A  /\  E. x  e.  y  ph ) )
3837exlimiv 1689 . 2  |-  ( E. z  z  e.  {
x  |  ( x  e.  A  /\  ph ) }  ->  E. y
( y  C_  A  /\  E. x  e.  y 
ph ) )
395, 38sylbi 195 1  |-  ( E. x  e.  A  ph  ->  E. y ( y 
C_  A  /\  E. x  e.  y  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587   [wsb 1702    e. wcel 1758   {cab 2436    =/= wne 2644   E.wrex 2796   {crab 2799    C_ wss 3429   (/)c0 3738   {csn 3978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-sn 3979  df-pr 3981
This theorem is referenced by: (None)
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