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Theorem php5dom 6356
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 4167 . . . 4  |-  ( w  =  (/)  ->  suc  w  =  suc  (/) )
2 id 19 . . . 4  |-  ( w  =  (/)  ->  w  =  (/) )
31, 2breq12d 3805 . . 3  |-  ( w  =  (/)  ->  ( suc  w  ~<_  w  <->  suc  (/)  ~<_  (/) ) )
43notbid 602 . 2  |-  ( w  =  (/)  ->  ( -. 
suc  w  ~<_  w  <->  -.  suc  (/)  ~<_  (/) ) )
5 suceq 4167 . . . 4  |-  ( w  =  k  ->  suc  w  =  suc  k )
6 id 19 . . . 4  |-  ( w  =  k  ->  w  =  k )
75, 6breq12d 3805 . . 3  |-  ( w  =  k  ->  ( suc  w  ~<_  w  <->  suc  k  ~<_  k ) )
87notbid 602 . 2  |-  ( w  =  k  ->  ( -.  suc  w  ~<_  w  <->  -.  suc  k  ~<_  k ) )
9 suceq 4167 . . . 4  |-  ( w  =  suc  k  ->  suc  w  =  suc  suc  k )
10 id 19 . . . 4  |-  ( w  =  suc  k  ->  w  =  suc  k )
119, 10breq12d 3805 . . 3  |-  ( w  =  suc  k  -> 
( suc  w  ~<_  w  <->  suc  suc  k  ~<_  suc  k ) )
1211notbid 602 . 2  |-  ( w  =  suc  k  -> 
( -.  suc  w  ~<_  w 
<->  -.  suc  suc  k  ~<_  suc  k ) )
13 suceq 4167 . . . 4  |-  ( w  =  A  ->  suc  w  =  suc  A )
14 id 19 . . . 4  |-  ( w  =  A  ->  w  =  A )
1513, 14breq12d 3805 . . 3  |-  ( w  =  A  ->  ( suc  w  ~<_  w  <->  suc  A  ~<_  A ) )
1615notbid 602 . 2  |-  ( w  =  A  ->  ( -.  suc  w  ~<_  w  <->  -.  suc  A  ~<_  A ) )
17 peano1 4345 . . . 4  |-  (/)  e.  om
18 php5 6352 . . . 4  |-  ( (/)  e.  om  ->  -.  (/)  ~~  suc  (/) )
1917, 18ax-mp 7 . . 3  |-  -.  (/)  ~~  suc  (/)
20 0ex 3912 . . . . . 6  |-  (/)  e.  _V
2120domen 6263 . . . . 5  |-  ( suc  (/) 
~<_  (/)  <->  E. x ( suc  (/)  ~~  x  /\  x  C_  (/) ) )
22 ss0 3285 . . . . . . . 8  |-  ( x 
C_  (/)  ->  x  =  (/) )
23 en0 6306 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
2422, 23sylibr 141 . . . . . . 7  |-  ( x 
C_  (/)  ->  x  ~~  (/) )
25 entr 6295 . . . . . . 7  |-  ( ( suc  (/)  ~~  x  /\  x  ~~  (/) )  ->  suc  (/)  ~~  (/) )
2624, 25sylan2 274 . . . . . 6  |-  ( ( suc  (/)  ~~  x  /\  x  C_  (/) )  ->  suc  (/)  ~~  (/) )
2726exlimiv 1505 . . . . 5  |-  ( E. x ( suc  (/)  ~~  x  /\  x  C_  (/) )  ->  suc  (/)  ~~  (/) )
2821, 27sylbi 118 . . . 4  |-  ( suc  (/) 
~<_  (/)  ->  suc  (/)  ~~  (/) )
2928ensymd 6294 . . 3  |-  ( suc  (/) 
~<_  (/)  ->  (/)  ~~  suc  (/) )
3019, 29mto 598 . 2  |-  -.  suc  (/)  ~<_  (/)
31 peano2 4346 . . . 4  |-  ( k  e.  om  ->  suc  k  e.  om )
32 phplem4dom 6355 . . . 4  |-  ( ( suc  k  e.  om  /\  k  e.  om )  ->  ( suc  suc  k  ~<_  suc  k  ->  suc  k  ~<_  k ) )
3331, 32mpancom 407 . . 3  |-  ( k  e.  om  ->  ( suc  suc  k  ~<_  suc  k  ->  suc  k  ~<_  k ) )
3433con3d 571 . 2  |-  ( k  e.  om  ->  ( -.  suc  k  ~<_  k  ->  -.  suc  suc  k  ~<_  suc  k
) )
354, 8, 12, 16, 30, 34finds 4351 1  |-  ( A  e.  om  ->  -.  suc  A  ~<_  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409    C_ wss 2945   (/)c0 3252   class class class wbr 3792   suc csuc 4130   omcom 4341    ~~ cen 6250    ~<_ cdom 6251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-er 6137  df-en 6253  df-dom 6254
This theorem is referenced by:  nndomo  6357  phpm  6358
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