Theorem php5dom 6708

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 4282	. . . 4 |- ( w = (/) -> suc w = suc (/) )
2		id 19	. . . 4 |- ( w = (/) -> w = (/) )
3	1, 2	breq12d 3906	. . 3 |- ( w = (/) -> ( suc w ~<_ w <-> suc (/) ~<_ (/) ) )
4	3	notbid 639	. 2 |- ( w = (/) -> ( -. suc w ~<_ w <-> -. suc (/) ~<_ (/) ) )
5		suceq 4282	. . . 4 |- ( w = k -> suc w = suc k )
6		id 19	. . . 4 |- ( w = k -> w = k )
7	5, 6	breq12d 3906	. . 3 |- ( w = k -> ( suc w ~<_ w <-> suc k ~<_ k ) )
8	7	notbid 639	. 2 |- ( w = k -> ( -. suc w ~<_ w <-> -. suc k ~<_ k ) )
9		suceq 4282	. . . 4 |- ( w = suc k -> suc w = suc suc k )
10		id 19	. . . 4 |- ( w = suc k -> w = suc k )
11	9, 10	breq12d 3906	. . 3 |- ( w = suc k -> ( suc w ~<_ w <-> suc suc k ~<_ suc k ) )
12	11	notbid 639	. 2 |- ( w = suc k -> ( -. suc w ~<_ w <-> -. suc suc k ~<_ suc k ) )
13		suceq 4282	. . . 4 |- ( w = A -> suc w = suc A )
14		id 19	. . . 4 |- ( w = A -> w = A )
15	13, 14	breq12d 3906	. . 3 |- ( w = A -> ( suc w ~<_ w <-> suc A ~<_ A ) )
16	15	notbid 639	. 2 |- ( w = A -> ( -. suc w ~<_ w <-> -. suc A ~<_ A ) )
17		peano1 4466	. . . 4 |- (/) e. om
18		php5 6703	. . . 4 |- ( (/) e. om -> -. (/) ~~ suc (/) )
19	17, 18	ax-mp 7	. . 3 |- -. (/) ~~ suc (/)
20		0ex 4013	. . . . . 6 |- (/) e. _V
21	20	domen 6597	. . . . 5 |- ( suc (/) ~<_ (/) <-> E. x ( suc (/) ~~ x /\ x C_ (/) ) )
22		ss0 3367	. . . . . . . 8 |- ( x C_ (/) -> x = (/) )
23		en0 6641	. . . . . . . 8 |- ( x ~~ (/) <-> x = (/) )
24	22, 23	sylibr 133	. . . . . . 7 |- ( x C_ (/) -> x ~~ (/) )
25		entr 6630	. . . . . . 7 |- ( ( suc (/) ~~ x /\ x ~~ (/) ) -> suc (/) ~~ (/) )
26	24, 25	sylan2 282	. . . . . 6 |- ( ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
27	26	exlimiv 1558	. . . . 5 |- ( E. x ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
28	21, 27	sylbi 120	. . . 4 |- ( suc (/) ~<_ (/) -> suc (/) ~~ (/) )
29	28	ensymd 6629	. . 3 |- ( suc (/) ~<_ (/) -> (/) ~~ suc (/) )
30	19, 29	mto 634	. 2 |- -. suc (/) ~<_ (/)
31		peano2 4467	. . . 4 |- ( k e. om -> suc k e. om )
32		phplem4dom 6707	. . . 4 |- ( ( suc k e. om /\ k e. om ) -> ( suc suc k ~<_ suc k -> suc k ~<_ k ) )
33	31, 32	mpancom 416	. . 3 |- ( k e. om -> ( suc suc k ~<_ suc k -> suc k ~<_ k ) )
34	33	con3d 603	. 2 |- ( k e. om -> ( -. suc k ~<_ k -> -. suc suc k ~<_ suc k ) )
35	4, 8, 12, 16, 30, 34	finds 4472	1 |- ( A e. om -> -. suc A ~<_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1312   E.wex 1449   e. wcel 1461   C_ wss 3035   (/)c0 3327   class class class wbr 3893   suc csuc 4245   omcom 4462   ~~ cen 6584   ~<_ cdom 6585

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-er 6381  df-en 6587  df-dom 6588

This theorem is referenced by:  nndomo  6709  phpm  6710  infnfi  6740