Theorem php5dom 6863

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 4403	. . . 4 |- ( w = (/) -> suc w = suc (/) )
2		id 19	. . . 4 |- ( w = (/) -> w = (/) )
3	1, 2	breq12d 4017	. . 3 |- ( w = (/) -> ( suc w ~<_ w <-> suc (/) ~<_ (/) ) )
4	3	notbid 667	. 2 |- ( w = (/) -> ( -. suc w ~<_ w <-> -. suc (/) ~<_ (/) ) )
5		suceq 4403	. . . 4 |- ( w = k -> suc w = suc k )
6		id 19	. . . 4 |- ( w = k -> w = k )
7	5, 6	breq12d 4017	. . 3 |- ( w = k -> ( suc w ~<_ w <-> suc k ~<_ k ) )
8	7	notbid 667	. 2 |- ( w = k -> ( -. suc w ~<_ w <-> -. suc k ~<_ k ) )
9		suceq 4403	. . . 4 |- ( w = suc k -> suc w = suc suc k )
10		id 19	. . . 4 |- ( w = suc k -> w = suc k )
11	9, 10	breq12d 4017	. . 3 |- ( w = suc k -> ( suc w ~<_ w <-> suc suc k ~<_ suc k ) )
12	11	notbid 667	. 2 |- ( w = suc k -> ( -. suc w ~<_ w <-> -. suc suc k ~<_ suc k ) )
13		suceq 4403	. . . 4 |- ( w = A -> suc w = suc A )
14		id 19	. . . 4 |- ( w = A -> w = A )
15	13, 14	breq12d 4017	. . 3 |- ( w = A -> ( suc w ~<_ w <-> suc A ~<_ A ) )
16	15	notbid 667	. 2 |- ( w = A -> ( -. suc w ~<_ w <-> -. suc A ~<_ A ) )
17		peano1 4594	. . . 4 |- (/) e. om
18		php5 6858	. . . 4 |- ( (/) e. om -> -. (/) ~~ suc (/) )
19	17, 18	ax-mp 5	. . 3 |- -. (/) ~~ suc (/)
20		0ex 4131	. . . . . 6 |- (/) e. _V
21	20	domen 6751	. . . . 5 |- ( suc (/) ~<_ (/) <-> E. x ( suc (/) ~~ x /\ x C_ (/) ) )
22		ss0 3464	. . . . . . . 8 |- ( x C_ (/) -> x = (/) )
23		en0 6795	. . . . . . . 8 |- ( x ~~ (/) <-> x = (/) )
24	22, 23	sylibr 134	. . . . . . 7 |- ( x C_ (/) -> x ~~ (/) )
25		entr 6784	. . . . . . 7 |- ( ( suc (/) ~~ x /\ x ~~ (/) ) -> suc (/) ~~ (/) )
26	24, 25	sylan2 286	. . . . . 6 |- ( ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
27	26	exlimiv 1598	. . . . 5 |- ( E. x ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
28	21, 27	sylbi 121	. . . 4 |- ( suc (/) ~<_ (/) -> suc (/) ~~ (/) )
29	28	ensymd 6783	. . 3 |- ( suc (/) ~<_ (/) -> (/) ~~ suc (/) )
30	19, 29	mto 662	. 2 |- -. suc (/) ~<_ (/)
31		peano2 4595	. . . 4 |- ( k e. om -> suc k e. om )
32		phplem4dom 6862	. . . 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 631	. 2 |- ( k e. om -> ( -. suc k ~<_ k -> -. suc suc k ~<_ suc k ) )
35	4, 8, 12, 16, 30, 34	finds 4600	1 |- ( A e. om -> -. suc A ~<_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   C_ wss 3130   (/)c0 3423   class class class wbr 4004   suc csuc 4366   omcom 4590   ~~ cen 6738   ~<_ cdom 6739

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 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-er 6535  df-en 6741  df-dom 6742

This theorem is referenced by:  nndomo  6864  phpm  6865  infnfi  6895