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Theorem php5dom 6559
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 4220 . . . 4  |-  ( w  =  (/)  ->  suc  w  =  suc  (/) )
2 id 19 . . . 4  |-  ( w  =  (/)  ->  w  =  (/) )
31, 2breq12d 3850 . . 3  |-  ( w  =  (/)  ->  ( suc  w  ~<_  w  <->  suc  (/)  ~<_  (/) ) )
43notbid 627 . 2  |-  ( w  =  (/)  ->  ( -. 
suc  w  ~<_  w  <->  -.  suc  (/)  ~<_  (/) ) )
5 suceq 4220 . . . 4  |-  ( w  =  k  ->  suc  w  =  suc  k )
6 id 19 . . . 4  |-  ( w  =  k  ->  w  =  k )
75, 6breq12d 3850 . . 3  |-  ( w  =  k  ->  ( suc  w  ~<_  w  <->  suc  k  ~<_  k ) )
87notbid 627 . 2  |-  ( w  =  k  ->  ( -.  suc  w  ~<_  w  <->  -.  suc  k  ~<_  k ) )
9 suceq 4220 . . . 4  |-  ( w  =  suc  k  ->  suc  w  =  suc  suc  k )
10 id 19 . . . 4  |-  ( w  =  suc  k  ->  w  =  suc  k )
119, 10breq12d 3850 . . 3  |-  ( w  =  suc  k  -> 
( suc  w  ~<_  w  <->  suc  suc  k  ~<_  suc  k ) )
1211notbid 627 . 2  |-  ( w  =  suc  k  -> 
( -.  suc  w  ~<_  w 
<->  -.  suc  suc  k  ~<_  suc  k ) )
13 suceq 4220 . . . 4  |-  ( w  =  A  ->  suc  w  =  suc  A )
14 id 19 . . . 4  |-  ( w  =  A  ->  w  =  A )
1513, 14breq12d 3850 . . 3  |-  ( w  =  A  ->  ( suc  w  ~<_  w  <->  suc  A  ~<_  A ) )
1615notbid 627 . 2  |-  ( w  =  A  ->  ( -.  suc  w  ~<_  w  <->  -.  suc  A  ~<_  A ) )
17 peano1 4399 . . . 4  |-  (/)  e.  om
18 php5 6554 . . . 4  |-  ( (/)  e.  om  ->  -.  (/)  ~~  suc  (/) )
1917, 18ax-mp 7 . . 3  |-  -.  (/)  ~~  suc  (/)
20 0ex 3958 . . . . . 6  |-  (/)  e.  _V
2120domen 6448 . . . . 5  |-  ( suc  (/) 
~<_  (/)  <->  E. x ( suc  (/)  ~~  x  /\  x  C_  (/) ) )
22 ss0 3320 . . . . . . . 8  |-  ( x 
C_  (/)  ->  x  =  (/) )
23 en0 6492 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
2422, 23sylibr 132 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  ~~  (/) )
25 entr 6481 . . . . . . 7  |-  ( ( suc  (/)  ~~  x  /\  x  ~~  (/) )  ->  suc  (/)  ~~  (/) )
2624, 25sylan2 280 . . . . . 6  |-  ( ( suc  (/)  ~~  x  /\  x  C_  (/) )  ->  suc  (/)  ~~  (/) )
2726exlimiv 1534 . . . . 5  |-  ( E. x ( suc  (/)  ~~  x  /\  x  C_  (/) )  ->  suc  (/)  ~~  (/) )
2821, 27sylbi 119 . . . 4  |-  ( suc  (/) 
~<_  (/)  ->  suc  (/)  ~~  (/) )
2928ensymd 6480 . . 3  |-  ( suc  (/) 
~<_  (/)  ->  (/)  ~~  suc  (/) )
3019, 29mto 623 . 2  |-  -.  suc  (/)  ~<_  (/)
31 peano2 4400 . . . 4  |-  ( k  e.  om  ->  suc  k  e.  om )
32 phplem4dom 6558 . . . 4  |-  ( ( suc  k  e.  om  /\  k  e.  om )  ->  ( suc  suc  k  ~<_  suc  k  ->  suc  k  ~<_  k ) )
3331, 32mpancom 413 . . 3  |-  ( k  e.  om  ->  ( suc  suc  k  ~<_  suc  k  ->  suc  k  ~<_  k ) )
3433con3d 596 . 2  |-  ( k  e.  om  ->  ( -.  suc  k  ~<_  k  ->  -.  suc  suc  k  ~<_  suc  k
) )
354, 8, 12, 16, 30, 34finds 4405 1  |-  ( A  e.  om  ->  -.  suc  A  ~<_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438    C_ wss 2997   (/)c0 3284   class class class wbr 3837   suc csuc 4183   omcom 4395    ~~ cen 6435    ~<_ cdom 6436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-er 6272  df-en 6438  df-dom 6439
This theorem is referenced by:  nndomo  6560  phpm  6561  infnfi  6591
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