Theorem php5dom 7020

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 4492	. . . 4 |- ( w = (/) -> suc w = suc (/) )
2		id 19	. . . 4 |- ( w = (/) -> w = (/) )
3	1, 2	breq12d 4095	. . 3 |- ( w = (/) -> ( suc w ~<_ w <-> suc (/) ~<_ (/) ) )
4	3	notbid 671	. 2 |- ( w = (/) -> ( -. suc w ~<_ w <-> -. suc (/) ~<_ (/) ) )
5		suceq 4492	. . . 4 |- ( w = k -> suc w = suc k )
6		id 19	. . . 4 |- ( w = k -> w = k )
7	5, 6	breq12d 4095	. . 3 |- ( w = k -> ( suc w ~<_ w <-> suc k ~<_ k ) )
8	7	notbid 671	. 2 |- ( w = k -> ( -. suc w ~<_ w <-> -. suc k ~<_ k ) )
9		suceq 4492	. . . 4 |- ( w = suc k -> suc w = suc suc k )
10		id 19	. . . 4 |- ( w = suc k -> w = suc k )
11	9, 10	breq12d 4095	. . 3 |- ( w = suc k -> ( suc w ~<_ w <-> suc suc k ~<_ suc k ) )
12	11	notbid 671	. 2 |- ( w = suc k -> ( -. suc w ~<_ w <-> -. suc suc k ~<_ suc k ) )
13		suceq 4492	. . . 4 |- ( w = A -> suc w = suc A )
14		id 19	. . . 4 |- ( w = A -> w = A )
15	13, 14	breq12d 4095	. . 3 |- ( w = A -> ( suc w ~<_ w <-> suc A ~<_ A ) )
16	15	notbid 671	. 2 |- ( w = A -> ( -. suc w ~<_ w <-> -. suc A ~<_ A ) )
17		peano1 4685	. . . 4 |- (/) e. om
18		php5 7015	. . . 4 |- ( (/) e. om -> -. (/) ~~ suc (/) )
19	17, 18	ax-mp 5	. . 3 |- -. (/) ~~ suc (/)
20		0ex 4210	. . . . . 6 |- (/) e. _V
21	20	domen 6898	. . . . 5 |- ( suc (/) ~<_ (/) <-> E. x ( suc (/) ~~ x /\ x C_ (/) ) )
22		ss0 3532	. . . . . . . 8 |- ( x C_ (/) -> x = (/) )
23		en0 6945	. . . . . . . 8 |- ( x ~~ (/) <-> x = (/) )
24	22, 23	sylibr 134	. . . . . . 7 |- ( x C_ (/) -> x ~~ (/) )
25		entr 6934	. . . . . . 7 |- ( ( suc (/) ~~ x /\ x ~~ (/) ) -> suc (/) ~~ (/) )
26	24, 25	sylan2 286	. . . . . 6 |- ( ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
27	26	exlimiv 1644	. . . . 5 |- ( E. x ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
28	21, 27	sylbi 121	. . . 4 |- ( suc (/) ~<_ (/) -> suc (/) ~~ (/) )
29	28	ensymd 6933	. . 3 |- ( suc (/) ~<_ (/) -> (/) ~~ suc (/) )
30	19, 29	mto 666	. 2 |- -. suc (/) ~<_ (/)
31		peano2 4686	. . . 4 |- ( k e. om -> suc k e. om )
32		phplem4dom 7019	. . . 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 634	. 2 |- ( k e. om -> ( -. suc k ~<_ k -> -. suc suc k ~<_ suc k ) )
35	4, 8, 12, 16, 30, 34	finds 4691	1 |- ( A e. om -> -. suc A ~<_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   C_ wss 3197   (/)c0 3491   class class class wbr 4082   suc csuc 4455   omcom 4681   ~~ cen 6883   ~<_ cdom 6884

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886  df-dom 6887

This theorem is referenced by:  nndomo  7021  phpm  7023  infnfi  7053