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Theorem php5 6872
Description: A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
Assertion
Ref Expression
php5  |-  ( A  e.  om  ->  -.  A  ~~  suc  A )

Proof of Theorem php5
Dummy variables  w  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4  |-  ( w  =  (/)  ->  w  =  (/) )
2 suceq 4414 . . . 4  |-  ( w  =  (/)  ->  suc  w  =  suc  (/) )
31, 2breq12d 4028 . . 3  |-  ( w  =  (/)  ->  ( w 
~~  suc  w  <->  (/)  ~~  suc  (/) ) )
43notbid 668 . 2  |-  ( w  =  (/)  ->  ( -.  w  ~~  suc  w  <->  -.  (/)  ~~  suc  (/) ) )
5 id 19 . . . 4  |-  ( w  =  k  ->  w  =  k )
6 suceq 4414 . . . 4  |-  ( w  =  k  ->  suc  w  =  suc  k )
75, 6breq12d 4028 . . 3  |-  ( w  =  k  ->  (
w  ~~  suc  w  <->  k  ~~  suc  k ) )
87notbid 668 . 2  |-  ( w  =  k  ->  ( -.  w  ~~  suc  w  <->  -.  k  ~~  suc  k
) )
9 id 19 . . . 4  |-  ( w  =  suc  k  ->  w  =  suc  k )
10 suceq 4414 . . . 4  |-  ( w  =  suc  k  ->  suc  w  =  suc  suc  k )
119, 10breq12d 4028 . . 3  |-  ( w  =  suc  k  -> 
( w  ~~  suc  w 
<->  suc  k  ~~  suc  suc  k ) )
1211notbid 668 . 2  |-  ( w  =  suc  k  -> 
( -.  w  ~~  suc  w  <->  -.  suc  k  ~~  suc  suc  k ) )
13 id 19 . . . 4  |-  ( w  =  A  ->  w  =  A )
14 suceq 4414 . . . 4  |-  ( w  =  A  ->  suc  w  =  suc  A )
1513, 14breq12d 4028 . . 3  |-  ( w  =  A  ->  (
w  ~~  suc  w  <->  A  ~~  suc  A ) )
1615notbid 668 . 2  |-  ( w  =  A  ->  ( -.  w  ~~  suc  w  <->  -.  A  ~~  suc  A
) )
17 peano1 4605 . . . . 5  |-  (/)  e.  om
18 peano3 4607 . . . . 5  |-  ( (/)  e.  om  ->  suc  (/)  =/=  (/) )
1917, 18ax-mp 5 . . . 4  |-  suc  (/)  =/=  (/)
20 en0 6809 . . . 4  |-  ( suc  (/)  ~~  (/)  <->  suc  (/)  =  (/) )
2119, 20nemtbir 2446 . . 3  |-  -.  suc  (/)  ~~  (/)
22 ensymb 6794 . . 3  |-  ( suc  (/)  ~~  (/)  <->  (/)  ~~  suc  (/) )
2321, 22mtbi 671 . 2  |-  -.  (/)  ~~  suc  (/)
24 peano2 4606 . . . 4  |-  ( k  e.  om  ->  suc  k  e.  om )
25 vex 2752 . . . . 5  |-  k  e. 
_V
2625sucex 4510 . . . . 5  |-  suc  k  e.  _V
2725, 26phplem4 6869 . . . 4  |-  ( ( k  e.  om  /\  suc  k  e.  om )  ->  ( suc  k  ~~  suc  suc  k  ->  k 
~~  suc  k )
)
2824, 27mpdan 421 . . 3  |-  ( k  e.  om  ->  ( suc  k  ~~  suc  suc  k  ->  k  ~~  suc  k ) )
2928con3d 632 . 2  |-  ( k  e.  om  ->  ( -.  k  ~~  suc  k  ->  -.  suc  k  ~~  suc  suc  k ) )
304, 8, 12, 16, 23, 29finds 4611 1  |-  ( A  e.  om  ->  -.  A  ~~  suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1363    e. wcel 2158    =/= wne 2357   (/)c0 3434   class class class wbr 4015   suc csuc 4377   omcom 4601    ~~ cen 6752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-er 6549  df-en 6755
This theorem is referenced by:  snnen2og  6873  1nen2  6875  php5dom  6877  php5fin  6896
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