Theorem php5dom 6980

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 4462	. . . 4 |- ( w = (/) -> suc w = suc (/) )
2		id 19	. . . 4 |- ( w = (/) -> w = (/) )
3	1, 2	breq12d 4067	. . 3 |- ( w = (/) -> ( suc w ~<_ w <-> suc (/) ~<_ (/) ) )
4	3	notbid 669	. 2 |- ( w = (/) -> ( -. suc w ~<_ w <-> -. suc (/) ~<_ (/) ) )
5		suceq 4462	. . . 4 |- ( w = k -> suc w = suc k )
6		id 19	. . . 4 |- ( w = k -> w = k )
7	5, 6	breq12d 4067	. . 3 |- ( w = k -> ( suc w ~<_ w <-> suc k ~<_ k ) )
8	7	notbid 669	. 2 |- ( w = k -> ( -. suc w ~<_ w <-> -. suc k ~<_ k ) )
9		suceq 4462	. . . 4 |- ( w = suc k -> suc w = suc suc k )
10		id 19	. . . 4 |- ( w = suc k -> w = suc k )
11	9, 10	breq12d 4067	. . 3 |- ( w = suc k -> ( suc w ~<_ w <-> suc suc k ~<_ suc k ) )
12	11	notbid 669	. 2 |- ( w = suc k -> ( -. suc w ~<_ w <-> -. suc suc k ~<_ suc k ) )
13		suceq 4462	. . . 4 |- ( w = A -> suc w = suc A )
14		id 19	. . . 4 |- ( w = A -> w = A )
15	13, 14	breq12d 4067	. . 3 |- ( w = A -> ( suc w ~<_ w <-> suc A ~<_ A ) )
16	15	notbid 669	. 2 |- ( w = A -> ( -. suc w ~<_ w <-> -. suc A ~<_ A ) )
17		peano1 4655	. . . 4 |- (/) e. om
18		php5 6975	. . . 4 |- ( (/) e. om -> -. (/) ~~ suc (/) )
19	17, 18	ax-mp 5	. . 3 |- -. (/) ~~ suc (/)
20		0ex 4182	. . . . . 6 |- (/) e. _V
21	20	domen 6858	. . . . 5 |- ( suc (/) ~<_ (/) <-> E. x ( suc (/) ~~ x /\ x C_ (/) ) )
22		ss0 3505	. . . . . . . 8 |- ( x C_ (/) -> x = (/) )
23		en0 6905	. . . . . . . 8 |- ( x ~~ (/) <-> x = (/) )
24	22, 23	sylibr 134	. . . . . . 7 |- ( x C_ (/) -> x ~~ (/) )
25		entr 6894	. . . . . . 7 |- ( ( suc (/) ~~ x /\ x ~~ (/) ) -> suc (/) ~~ (/) )
26	24, 25	sylan2 286	. . . . . 6 |- ( ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
27	26	exlimiv 1622	. . . . 5 |- ( E. x ( suc (/) ~~ x /\ x C_ (/) ) -> suc (/) ~~ (/) )
28	21, 27	sylbi 121	. . . 4 |- ( suc (/) ~<_ (/) -> suc (/) ~~ (/) )
29	28	ensymd 6893	. . 3 |- ( suc (/) ~<_ (/) -> (/) ~~ suc (/) )
30	19, 29	mto 664	. 2 |- -. suc (/) ~<_ (/)
31		peano2 4656	. . . 4 |- ( k e. om -> suc k e. om )
32		phplem4dom 6979	. . . 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 632	. 2 |- ( k e. om -> ( -. suc k ~<_ k -> -. suc suc k ~<_ suc k ) )
35	4, 8, 12, 16, 30, 34	finds 4661	1 |- ( A e. om -> -. suc A ~<_ A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2177   C_ wss 3170   (/)c0 3464   class class class wbr 4054   suc csuc 4425   omcom 4651   ~~ cen 6843   ~<_ cdom 6844

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-er 6638  df-en 6846  df-dom 6847

This theorem is referenced by:  nndomo  6981  phpm  6983  infnfi  7013