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Theorem enp1i 7336
Description: Proof induction for en2i 7138 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 4654 . . . . 5  |-  suc  M  =/=  (/)
2 breq1 4208 . . . . . . 7  |-  ( A  =  (/)  ->  ( A 
~~  N  <->  (/)  ~~  N
) )
3 enp1i.2 . . . . . . . 8  |-  N  =  suc  M
4 ensym 7149 . . . . . . . . 9  |-  ( (/)  ~~  N  ->  N  ~~  (/) )
5 en0 7163 . . . . . . . . 9  |-  ( N 
~~  (/)  <->  N  =  (/) )
64, 5sylib 189 . . . . . . . 8  |-  ( (/)  ~~  N  ->  N  =  (/) )
73, 6syl5eqr 2482 . . . . . . 7  |-  ( (/)  ~~  N  ->  suc  M  =  (/) )
82, 7syl6bi 220 . . . . . 6  |-  ( A  =  (/)  ->  ( A 
~~  N  ->  suc  M  =  (/) ) )
98necon3ad 2635 . . . . 5  |-  ( A  =  (/)  ->  ( suc 
M  =/=  (/)  ->  -.  A  ~~  N ) )
101, 9mpi 17 . . . 4  |-  ( A  =  (/)  ->  -.  A  ~~  N )
1110con2i 114 . . 3  |-  ( A 
~~  N  ->  -.  A  =  (/) )
12 neq0 3631 . . 3  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
1311, 12sylib 189 . 2  |-  ( A 
~~  N  ->  E. x  x  e.  A )
143breq2i 4213 . . . . 5  |-  ( A 
~~  N  <->  A  ~~  suc  M )
15 enp1i.1 . . . . . . . 8  |-  M  e. 
om
16 dif1en 7334 . . . . . . . 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 1632 . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3310   (/)c0 3621   {csn 3807   class class class wbr 4205   suc csuc 4576   omcom 4838    ~~ cen 7099
This theorem is referenced by:  en2  7337  en3  7338  en4  7339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-1o 6717  df-er 6898  df-en 7103  df-fin 7106
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