MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  enp1i Structured version   Unicode version

Theorem enp1i 7670
Description: Proof induction for en2i 7472 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 4872 . . . . 5  |-  suc  M  =/=  (/)
2 breq1 4370 . . . . . . 7  |-  ( A  =  (/)  ->  ( A 
~~  N  <->  (/)  ~~  N
) )
3 enp1i.2 . . . . . . . 8  |-  N  =  suc  M
4 ensym 7483 . . . . . . . . 9  |-  ( (/)  ~~  N  ->  N  ~~  (/) )
5 en0 7497 . . . . . . . . 9  |-  ( N 
~~  (/)  <->  N  =  (/) )
64, 5sylib 196 . . . . . . . 8  |-  ( (/)  ~~  N  ->  N  =  (/) )
73, 6syl5eqr 2437 . . . . . . 7  |-  ( (/)  ~~  N  ->  suc  M  =  (/) )
82, 7syl6bi 228 . . . . . 6  |-  ( A  =  (/)  ->  ( A 
~~  N  ->  suc  M  =  (/) ) )
98necon3ad 2592 . . . . 5  |-  ( A  =  (/)  ->  ( suc 
M  =/=  (/)  ->  -.  A  ~~  N ) )
101, 9mpi 17 . . . 4  |-  ( A  =  (/)  ->  -.  A  ~~  N )
1110con2i 120 . . 3  |-  ( A 
~~  N  ->  -.  A  =  (/) )
12 neq0 3722 . . 3  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
1311, 12sylib 196 . 2  |-  ( A 
~~  N  ->  E. x  x  e.  A )
143breq2i 4375 . . . . 5  |-  ( A 
~~  N  <->  A  ~~  suc  M )
15 enp1i.1 . . . . . . . 8  |-  M  e. 
om
16 dif1en 7668 . . . . . . . 8  |-  ( ( M  e.  om  /\  A  ~~  suc  M  /\  x  e.  A )  ->  ( A  \  {
x } )  ~~  M )
1715, 16mp3an1 1309 . . . . . . 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 432 . . . . 5  |-  ( A 
~~  suc  M  ->  ( x  e.  A  ->  ph ) )
2114, 20sylbi 195 . . . 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 1718 . 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 367    = wceq 1399   E.wex 1620    e. wcel 1826    =/= wne 2577    \ cdif 3386   (/)c0 3711   {csn 3944   class class class wbr 4367   suc csuc 4794   omcom 6599    ~~ cen 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-1o 7048  df-er 7229  df-en 7436  df-fin 7439
This theorem is referenced by:  en2  7671  en3  7672  en4  7673
  Copyright terms: Public domain W3C validator