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Theorem enp1i 7824
Description: Proof induction for en2i 7625 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.)
Hypotheses
Ref Expression
enp1i.1  |-  M  e. 
om
enp1i.2  |-  N  =  suc  M
enp1i.3  |-  ( ( A  \  { x } )  ~~  M  ->  ph )
enp1i.4  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
enp1i  |-  ( A 
~~  N  ->  E. x ps )
Distinct variable groups:    x, A    x, N
Allowed substitution hints:    ph( x)    ps( x)    M( x)

Proof of Theorem enp1i
StepHypRef Expression
1 nsuceq0 5510 . . . . 5  |-  suc  M  =/=  (/)
2 breq1 4398 . . . . . . 7  |-  ( A  =  (/)  ->  ( A 
~~  N  <->  (/)  ~~  N
) )
3 enp1i.2 . . . . . . . 8  |-  N  =  suc  M
4 ensym 7636 . . . . . . . . 9  |-  ( (/)  ~~  N  ->  N  ~~  (/) )
5 en0 7650 . . . . . . . . 9  |-  ( N 
~~  (/)  <->  N  =  (/) )
64, 5sylib 201 . . . . . . . 8  |-  ( (/)  ~~  N  ->  N  =  (/) )
73, 6syl5eqr 2519 . . . . . . 7  |-  ( (/)  ~~  N  ->  suc  M  =  (/) )
82, 7syl6bi 236 . . . . . 6  |-  ( A  =  (/)  ->  ( A 
~~  N  ->  suc  M  =  (/) ) )
98necon3ad 2656 . . . . 5  |-  ( A  =  (/)  ->  ( suc 
M  =/=  (/)  ->  -.  A  ~~  N ) )
101, 9mpi 20 . . . 4  |-  ( A  =  (/)  ->  -.  A  ~~  N )
1110con2i 124 . . 3  |-  ( A 
~~  N  ->  -.  A  =  (/) )
12 neq0 3733 . . 3  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
1311, 12sylib 201 . 2  |-  ( A 
~~  N  ->  E. x  x  e.  A )
143breq2i 4403 . . . . 5  |-  ( A 
~~  N  <->  A  ~~  suc  M )
15 enp1i.1 . . . . . . . 8  |-  M  e. 
om
16 dif1en 7822 . . . . . . . 8  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  x  e.  A )  ->  ( A  \  {
x } )  ~~  M )
1715, 16mp3an1 1377 . . . . . . 7  |-  ( ( A  ~~  suc  M  /\  x  e.  A
)  ->  ( A  \  { x } ) 
~~  M )
18 enp1i.3 . . . . . . 7  |-  ( ( A  \  { x } )  ~~  M  ->  ph )
1917, 18syl 17 . . . . . 6  |-  ( ( A  ~~  suc  M  /\  x  e.  A
)  ->  ph )
2019ex 441 . . . . 5  |-  ( A 
~~  suc  M  ->  ( x  e.  A  ->  ph ) )
2114, 20sylbi 200 . . . 4  |-  ( A 
~~  N  ->  (
x  e.  A  ->  ph ) )
22 enp1i.4 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
2321, 22sylcom 29 . . 3  |-  ( A 
~~  N  ->  (
x  e.  A  ->  ps ) )
2423eximdv 1772 . 2  |-  ( A 
~~  N  ->  ( E. x  x  e.  A  ->  E. x ps )
)
2513, 24mpd 15 1  |-  ( A 
~~  N  ->  E. x ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641    \ cdif 3387   (/)c0 3722   {csn 3959   class class class wbr 4395   suc csuc 5432   omcom 6711    ~~ cen 7584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-er 7381  df-en 7588  df-fin 7591
This theorem is referenced by:  en2  7825  en3  7826  en4  7827
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