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Theorem enp1i 7279
Description: Proof induction for en2i 7081 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 4602 . . . . 5  |-  suc  M  =/=  (/)
2 breq1 4156 . . . . . . 7  |-  ( A  =  (/)  ->  ( A 
~~  N  <->  (/)  ~~  N
) )
3 enp1i.2 . . . . . . . 8  |-  N  =  suc  M
4 ensym 7092 . . . . . . . . 9  |-  ( (/)  ~~  N  ->  N  ~~  (/) )
5 en0 7106 . . . . . . . . 9  |-  ( N 
~~  (/)  <->  N  =  (/) )
64, 5sylib 189 . . . . . . . 8  |-  ( (/)  ~~  N  ->  N  =  (/) )
73, 6syl5eqr 2433 . . . . . . 7  |-  ( (/)  ~~  N  ->  suc  M  =  (/) )
82, 7syl6bi 220 . . . . . 6  |-  ( A  =  (/)  ->  ( A 
~~  N  ->  suc  M  =  (/) ) )
98necon3ad 2586 . . . . 5  |-  ( A  =  (/)  ->  ( suc 
M  =/=  (/)  ->  -.  A  ~~  N ) )
101, 9mpi 17 . . . 4  |-  ( A  =  (/)  ->  -.  A  ~~  N )
1110con2i 114 . . 3  |-  ( A 
~~  N  ->  -.  A  =  (/) )
12 neq0 3581 . . 3  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
1311, 12sylib 189 . 2  |-  ( A 
~~  N  ->  E. x  x  e.  A )
143breq2i 4161 . . . . 5  |-  ( A 
~~  N  <->  A  ~~  suc  M )
15 enp1i.1 . . . . . . . 8  |-  M  e. 
om
16 dif1en 7277 . . . . . . . 8  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  x  e.  A )  ->  ( A  \  {
x } )  ~~  M )
1715, 16mp3an1 1266 . . . . . . 7  |-  ( ( A  ~~  suc  M  /\  x  e.  A
)  ->  ( A  \  { x } ) 
~~  M )
18 enp1i.3 . . . . . . 7  |-  ( ( A  \  { x } )  ~~  M  ->  ph )
1917, 18syl 16 . . . . . 6  |-  ( ( A  ~~  suc  M  /\  x  e.  A
)  ->  ph )
2019ex 424 . . . . 5  |-  ( A 
~~  suc  M  ->  ( x  e.  A  ->  ph ) )
2114, 20sylbi 188 . . . 4  |-  ( A 
~~  N  ->  (
x  e.  A  ->  ph ) )
22 enp1i.4 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
2321, 22sylcom 27 . . 3  |-  ( A 
~~  N  ->  (
x  e.  A  ->  ps ) )
2423eximdv 1629 . 2  |-  ( A 
~~  N  ->  ( E. x  x  e.  A  ->  E. x ps )
)
2513, 24mpd 15 1  |-  ( A 
~~  N  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2550    \ cdif 3260   (/)c0 3571   {csn 3757   class class class wbr 4153   suc csuc 4524   omcom 4785    ~~ cen 7042
This theorem is referenced by:  en2  7280  en3  7281  en4  7282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-1o 6660  df-er 6841  df-en 7046  df-fin 7049
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