ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  php5dom Unicode version

Theorem php5dom 6856
Description: A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
Assertion
Ref Expression
php5dom  |-  ( A  e.  om  ->  -.  suc  A  ~<_  A )

Proof of Theorem php5dom
Dummy variables  w  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suceq 4398 . . . 4  |-  ( w  =  (/)  ->  suc  w  =  suc  (/) )
2 id 19 . . . 4  |-  ( w  =  (/)  ->  w  =  (/) )
31, 2breq12d 4013 . . 3  |-  ( w  =  (/)  ->  ( suc  w  ~<_  w  <->  suc  (/)  ~<_  (/) ) )
43notbid 667 . 2  |-  ( w  =  (/)  ->  ( -. 
suc  w  ~<_  w  <->  -.  suc  (/)  ~<_  (/) ) )
5 suceq 4398 . . . 4  |-  ( w  =  k  ->  suc  w  =  suc  k )
6 id 19 . . . 4  |-  ( w  =  k  ->  w  =  k )
75, 6breq12d 4013 . . 3  |-  ( w  =  k  ->  ( suc  w  ~<_  w  <->  suc  k  ~<_  k ) )
87notbid 667 . 2  |-  ( w  =  k  ->  ( -.  suc  w  ~<_  w  <->  -.  suc  k  ~<_  k ) )
9 suceq 4398 . . . 4  |-  ( w  =  suc  k  ->  suc  w  =  suc  suc  k )
10 id 19 . . . 4  |-  ( w  =  suc  k  ->  w  =  suc  k )
119, 10breq12d 4013 . . 3  |-  ( w  =  suc  k  -> 
( suc  w  ~<_  w  <->  suc  suc  k  ~<_  suc  k ) )
1211notbid 667 . 2  |-  ( w  =  suc  k  -> 
( -.  suc  w  ~<_  w 
<->  -.  suc  suc  k  ~<_  suc  k ) )
13 suceq 4398 . . . 4  |-  ( w  =  A  ->  suc  w  =  suc  A )
14 id 19 . . . 4  |-  ( w  =  A  ->  w  =  A )
1513, 14breq12d 4013 . . 3  |-  ( w  =  A  ->  ( suc  w  ~<_  w  <->  suc  A  ~<_  A ) )
1615notbid 667 . 2  |-  ( w  =  A  ->  ( -.  suc  w  ~<_  w  <->  -.  suc  A  ~<_  A ) )
17 peano1 4589 . . . 4  |-  (/)  e.  om
18 php5 6851 . . . 4  |-  ( (/)  e.  om  ->  -.  (/)  ~~  suc  (/) )
1917, 18ax-mp 5 . . 3  |-  -.  (/)  ~~  suc  (/)
20 0ex 4127 . . . . . 6  |-  (/)  e.  _V
2120domen 6744 . . . . 5  |-  ( suc  (/) 
~<_  (/)  <->  E. x ( suc  (/)  ~~  x  /\  x  C_  (/) ) )
22 ss0 3463 . . . . . . . 8  |-  ( x 
C_  (/)  ->  x  =  (/) )
23 en0 6788 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
2422, 23sylibr 134 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  ~~  (/) )
25 entr 6777 . . . . . . 7  |-  ( ( suc  (/)  ~~  x  /\  x  ~~  (/) )  ->  suc  (/)  ~~  (/) )
2624, 25sylan2 286 . . . . . 6  |-  ( ( suc  (/)  ~~  x  /\  x  C_  (/) )  ->  suc  (/)  ~~  (/) )
2726exlimiv 1598 . . . . 5  |-  ( E. x ( suc  (/)  ~~  x  /\  x  C_  (/) )  ->  suc  (/)  ~~  (/) )
2821, 27sylbi 121 . . . 4  |-  ( suc  (/) 
~<_  (/)  ->  suc  (/)  ~~  (/) )
2928ensymd 6776 . . 3  |-  ( suc  (/) 
~<_  (/)  ->  (/)  ~~  suc  (/) )
3019, 29mto 662 . 2  |-  -.  suc  (/)  ~<_  (/)
31 peano2 4590 . . . 4  |-  ( k  e.  om  ->  suc  k  e.  om )
32 phplem4dom 6855 . . . 4  |-  ( ( suc  k  e.  om  /\  k  e.  om )  ->  ( suc  suc  k  ~<_  suc  k  ->  suc  k  ~<_  k ) )
3331, 32mpancom 422 . . 3  |-  ( k  e.  om  ->  ( suc  suc  k  ~<_  suc  k  ->  suc  k  ~<_  k ) )
3433con3d 631 . 2  |-  ( k  e.  om  ->  ( -.  suc  k  ~<_  k  ->  -.  suc  suc  k  ~<_  suc  k
) )
354, 8, 12, 16, 30, 34finds 4595 1  |-  ( A  e.  om  ->  -.  suc  A  ~<_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148    C_ wss 3129   (/)c0 3422   class class class wbr 4000   suc csuc 4361   omcom 4585    ~~ cen 6731    ~<_ cdom 6732
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-er 6528  df-en 6734  df-dom 6735
This theorem is referenced by:  nndomo  6857  phpm  6858  infnfi  6888
  Copyright terms: Public domain W3C validator