Theorem php5dom 7117

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.

Step	Hyp	Ref	Expression
1		suceq 4523	. . . 4 |- ( w = (/) -> suc w = suc (/) )
2		id 19	. . . 4 |- ( w = (/) -> w = (/) )
3	1, 2	breq12d 4122	. . 3 |- ( w = (/) -> ( suc w ~<_ w <-> suc (/) ~<_ (/) ) )
4	3	notbid 673	. 2 |- ( w = (/) -> ( -. suc w ~<_ w <-> -. suc (/) ~<_ (/) ) )
5		suceq 4523	. . . 4 |- ( w = k -> suc w = suc k )
6		id 19	. . . 4 |- ( w = k -> w = k )
7	5, 6	breq12d 4122	. . 3 |- ( w = k -> ( suc w ~<_ w <-> suc k ~<_ k ) )
8	7	notbid 673	. 2 |- ( w = k -> ( -. suc w ~<_ w <-> -. suc k ~<_ k ) )
9		suceq 4523	. . . 4 |- ( w = suc k -> suc w = suc suc k )
10		id 19	. . . 4 |- ( w = suc k -> w = suc k )
11	9, 10	breq12d 4122	. . 3 |- ( w = suc k -> ( suc w ~<_ w <-> suc suc k ~<_ suc k ) )
12	11	notbid 673	. 2 |- ( w = suc k -> ( -. suc w ~<_ w <-> -. suc suc k ~<_ suc k ) )
13		suceq 4523	. . . 4 |- ( w = A -> suc w = suc A )
14		id 19	. . . 4 |- ( w = A -> w = A )
15	13, 14	breq12d 4122	. . 3 |- ( w = A -> ( suc w ~<_ w <-> suc A ~<_ A ) )
16	15	notbid 673	. 2 |- ( w = A -> ( -. suc w ~<_ w <-> -. suc A ~<_ A ) )
17		peano1 4716	. . . 4 |- (/) e. om
18		php5 7112	. . . 4 |- ( (/) e. om -> -. (/) ~~ suc (/) )
19	17, 18	ax-mp 5	. . 3 |- -. (/) ~~ suc (/)
20		0ex 4237	. . . . . 6 |- (/) e. _V
21	20	domen 6988	. . . . 5 |- ( suc (/) ~<_ (/) <-> E. x ( suc (/) ~~ x /\ x C_ (/) ) )
22		ss0 3549	. . . . . . . 8 |- ( x C_ (/) -> x = (/) )
23		en0 7035	. . . . . . . 8 |- ( x ~~ (/) <-> x = (/) )
24	22, 23	sylibr 134	. . . . . . 7 |- ( x C_ (/) -> x ~~ (/) )
25		entr 7024	. . . . . . 7 |- ( ( suc (/) ~~ x /\ x ~~ (/) ) -> suc (/) ~~ (/) )
26	24, 25	sylan2 286	. . . . . 6 |- ( ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
27	26	exlimiv 1647	. . . . 5 |- ( E. x ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
28	21, 27	sylbi 121	. . . 4 |- ( suc (/) ~<_ (/) -> suc (/) ~~ (/) )
29	28	ensymd 7023	. . 3 |- ( suc (/) ~<_ (/) -> (/) ~~ suc (/) )
30	19, 29	mto 668	. 2 |- -. suc (/) ~<_ (/)
31		peano2 4717	. . . 4 |- ( k e. om -> suc k e. om )
32		phplem4dom 7116	. . . 4 |- ( ( suc k e. om /\ k e. om ) -> ( suc suc k ~<_ suc k -> suc k ~<_ k ) )
33	31, 32	mpancom 422	. . 3 |- ( k e. om -> ( suc suc k ~<_ suc k -> suc k ~<_ k ) )
34	33	con3d 636	. 2 |- ( k e. om -> ( -. suc k ~<_ k -> -. suc suc k ~<_ suc k ) )
35	4, 8, 12, 16, 30, 34	finds 4722	1 |- ( A e. om -> -. suc A ~<_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   C_ wss 3211   (/)c0 3508   class class class wbr 4109   suc csuc 4486   omcom 4712   ~~ cen 6973   ~<_ cdom 6974

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-er 6767  df-en 6976  df-dom 6977

This theorem is referenced by:  nndomo  7118  phpm  7120  infnfi  7152