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Theorem enp1i 7109
Description: Proof induction for en2i 6915 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 4488 . . . . 5  |-  suc  M  =/=  (/)
2 breq1 4042 . . . . . . 7  |-  ( A  =  (/)  ->  ( A 
~~  N  <->  (/)  ~~  N
) )
3 enp1i.2 . . . . . . . 8  |-  N  =  suc  M
4 ensym 6926 . . . . . . . . 9  |-  ( (/)  ~~  N  ->  N  ~~  (/) )
5 en0 6940 . . . . . . . . 9  |-  ( N 
~~  (/)  <->  N  =  (/) )
64, 5sylib 188 . . . . . . . 8  |-  ( (/)  ~~  N  ->  N  =  (/) )
73, 6syl5eqr 2342 . . . . . . 7  |-  ( (/)  ~~  N  ->  suc  M  =  (/) )
82, 7syl6bi 219 . . . . . 6  |-  ( A  =  (/)  ->  ( A 
~~  N  ->  suc  M  =  (/) ) )
98necon3ad 2495 . . . . 5  |-  ( A  =  (/)  ->  ( suc 
M  =/=  (/)  ->  -.  A  ~~  N ) )
101, 9mpi 16 . . . 4  |-  ( A  =  (/)  ->  -.  A  ~~  N )
1110con2i 112 . . 3  |-  ( A 
~~  N  ->  -.  A  =  (/) )
12 neq0 3478 . . 3  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
1311, 12sylib 188 . 2  |-  ( A 
~~  N  ->  E. x  x  e.  A )
143breq2i 4047 . . . . 5  |-  ( A 
~~  N  <->  A  ~~  suc  M )
15 enp1i.1 . . . . . . . 8  |-  M  e. 
om
16 dif1en 7107 . . . . . . . 8  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  x  e.  A )  ->  ( A  \  {
x } )  ~~  M )
1715, 16mp3an1 1264 . . . . . . 7  |-  ( ( A  ~~  suc  M  /\  x  e.  A
)  ->  ( A  \  { x } ) 
~~  M )
18 enp1i.3 . . . . . . 7  |-  ( ( A  \  { x } )  ~~  M  ->  ph )
1917, 18syl 15 . . . . . 6  |-  ( ( A  ~~  suc  M  /\  x  e.  A
)  ->  ph )
2019ex 423 . . . . 5  |-  ( A 
~~  suc  M  ->  ( x  e.  A  ->  ph ) )
2114, 20sylbi 187 . . . 4  |-  ( A 
~~  N  ->  (
x  e.  A  ->  ph ) )
22 enp1i.4 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
2321, 22sylcom 25 . . 3  |-  ( A 
~~  N  ->  (
x  e.  A  ->  ps ) )
2423eximdv 1612 . 2  |-  ( A 
~~  N  ->  ( E. x  x  e.  A  ->  E. x ps )
)
2513, 24mpd 14 1  |-  ( A 
~~  N  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   (/)c0 3468   {csn 3653   class class class wbr 4039   suc csuc 4410   omcom 4672    ~~ cen 6876
This theorem is referenced by:  en2  7110  en3  7111  en4  7112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-fin 6883
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