Theorem php5 6344

Description: A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)

Assertion

Ref	Expression
php5	|- ( A e. om -> -. A ~~ suc A )

Proof of Theorem php5

Dummy variables w k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- ( w = (/) -> w = (/) )
2		suceq 4157	. . . 4 |- ( w = (/) -> suc w = suc (/) )
3	1, 2	breq12d 3798	. . 3 |- ( w = (/) -> ( w ~~ suc w <-> (/) ~~ suc (/) ) )
4	3	notbid 624	. 2 |- ( w = (/) -> ( -. w ~~ suc w <-> -. (/) ~~ suc (/) ) )
5		id 19	. . . 4 |- ( w = k -> w = k )
6		suceq 4157	. . . 4 |- ( w = k -> suc w = suc k )
7	5, 6	breq12d 3798	. . 3 |- ( w = k -> ( w ~~ suc w <-> k ~~ suc k ) )
8	7	notbid 624	. 2 |- ( w = k -> ( -. w ~~ suc w <-> -. k ~~ suc k ) )
9		id 19	. . . 4 |- ( w = suc k -> w = suc k )
10		suceq 4157	. . . 4 |- ( w = suc k -> suc w = suc suc k )
11	9, 10	breq12d 3798	. . 3 |- ( w = suc k -> ( w ~~ suc w <-> suc k ~~ suc suc k ) )
12	11	notbid 624	. 2 |- ( w = suc k -> ( -. w ~~ suc w <-> -. suc k ~~ suc suc k ) )
13		id 19	. . . 4 |- ( w = A -> w = A )
14		suceq 4157	. . . 4 |- ( w = A -> suc w = suc A )
15	13, 14	breq12d 3798	. . 3 |- ( w = A -> ( w ~~ suc w <-> A ~~ suc A ) )
16	15	notbid 624	. 2 |- ( w = A -> ( -. w ~~ suc w <-> -. A ~~ suc A ) )
17		peano1 4335	. . . . 5 |- (/) e. om
18		peano3 4337	. . . . 5 |- ( (/) e. om -> suc (/) =/= (/) )
19	17, 18	ax-mp 7	. . . 4 |- suc (/) =/= (/)
20		en0 6298	. . . 4 |- ( suc (/) ~~ (/) <-> suc (/) = (/) )
21	19, 20	nemtbir 2334	. . 3 |- -. suc (/) ~~ (/)
22		ensymb 6283	. . 3 |- ( suc (/) ~~ (/) <-> (/) ~~ suc (/) )
23	21, 22	mtbi 627	. 2 |- -. (/) ~~ suc (/)
24		peano2 4336	. . . 4 |- ( k e. om -> suc k e. om )
25		vex 2604	. . . . 5 |- k e. _V
26	25	sucex 4243	. . . . 5 |- suc k e. _V
27	25, 26	phplem4 6341	. . . 4 |- ( ( k e. om /\ suc k e. om ) -> ( suc k ~~ suc suc k -> k ~~ suc k ) )
28	24, 27	mpdan 412	. . 3 |- ( k e. om -> ( suc k ~~ suc suc k -> k ~~ suc k ) )
29	28	con3d 593	. 2 |- ( k e. om -> ( -. k ~~ suc k -> -. suc k ~~ suc suc k ) )
30	4, 8, 12, 16, 23, 29	finds 4341	1 |- ( A e. om -> -. A ~~ suc A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1284   e. wcel 1433   =/= wne 2245   (/)c0 3251   class class class wbr 3785   suc csuc 4120   omcom 4331   ~~ cen 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-er 6129  df-en 6245

This theorem is referenced by:  snnen2og  6345  1nen2  6347  php5dom  6349  php5fin  6366