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Theorem exss 4719
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 2816 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21neeq1i 2742 . . 3  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
3 rabn0 3814 . . 3  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  E. x  e.  A  ph )
4 n0 3803 . . 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 3112 . . . . . 6  |-  z  e. 
_V
76snss 4156 . . . . 5  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  { z }  C_  { x  |  ( x  e.  A  /\  ph ) } )
8 ssab2 3580 . . . . . 6  |-  { x  |  ( x  e.  A  /\  ph ) }  C_  A
9 sstr2 3506 . . . . . 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 2108 . . . . . . . . 9  |-  [ z  /  x ] x  =  z
14 elsn 4046 . . . . . . . . . 10  |-  ( x  e.  { z }  <-> 
x  =  z )
1514sbbii 1747 . . . . . . . . 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 2443 . . . . . . . 8  |-  ( z  e.  { x  |  ( x  e.  A  /\  ph ) }  <->  [ z  /  x ] ( x  e.  A  /\  ph ) )
19 sban 2141 . . . . . . . 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 2816 . . . . . . . . 9  |-  { x  e.  { z }  |  ph }  =  { x  |  ( x  e. 
{ z }  /\  ph ) }
2221eleq2i 2535 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ z }  |  ph }  <->  z  e.  {
x  |  ( x  e.  { z }  /\  ph ) } )
23 df-clab 2443 . . . . . . . . 9  |-  ( z  e.  { x  |  ( x  e.  {
z }  /\  ph ) }  <->  [ z  /  x ] ( x  e. 
{ z }  /\  ph ) )
24 sban 2141 . . . . . . . . 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 3799 . . . . . 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 3814 . . . . 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 4697 . . . . 5  |-  { z }  e.  _V
33 sseq1 3520 . . . . . 6  |-  ( y  =  { z }  ->  ( y  C_  A 
<->  { z }  C_  A ) )
34 rexeq 3055 . . . . . 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 3201 . . . 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 1723 . 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 1395   E.wex 1613   [wsb 1740    e. wcel 1819   {cab 2442    =/= wne 2652   E.wrex 2808   {crab 2811    C_ wss 3471   (/)c0 3793   {csn 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-sn 4033  df-pr 4035
This theorem is referenced by: (None)
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