Theorem php5 6914

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 4433	. . . 4 |- ( w = (/) -> suc w = suc (/) )
3	1, 2	breq12d 4042	. . 3 |- ( w = (/) -> ( w ~~ suc w <-> (/) ~~ suc (/) ) )
4	3	notbid 668	. 2 |- ( w = (/) -> ( -. w ~~ suc w <-> -. (/) ~~ suc (/) ) )
5		id 19	. . . 4 |- ( w = k -> w = k )
6		suceq 4433	. . . 4 |- ( w = k -> suc w = suc k )
7	5, 6	breq12d 4042	. . 3 |- ( w = k -> ( w ~~ suc w <-> k ~~ suc k ) )
8	7	notbid 668	. 2 |- ( w = k -> ( -. w ~~ suc w <-> -. k ~~ suc k ) )
9		id 19	. . . 4 |- ( w = suc k -> w = suc k )
10		suceq 4433	. . . 4 |- ( w = suc k -> suc w = suc suc k )
11	9, 10	breq12d 4042	. . 3 |- ( w = suc k -> ( w ~~ suc w <-> suc k ~~ suc suc k ) )
12	11	notbid 668	. 2 |- ( w = suc k -> ( -. w ~~ suc w <-> -. suc k ~~ suc suc k ) )
13		id 19	. . . 4 |- ( w = A -> w = A )
14		suceq 4433	. . . 4 |- ( w = A -> suc w = suc A )
15	13, 14	breq12d 4042	. . 3 |- ( w = A -> ( w ~~ suc w <-> A ~~ suc A ) )
16	15	notbid 668	. 2 |- ( w = A -> ( -. w ~~ suc w <-> -. A ~~ suc A ) )
17		peano1 4626	. . . . 5 |- (/) e. om
18		peano3 4628	. . . . 5 |- ( (/) e. om -> suc (/) =/= (/) )
19	17, 18	ax-mp 5	. . . 4 |- suc (/) =/= (/)
20		en0 6849	. . . 4 |- ( suc (/) ~~ (/) <-> suc (/) = (/) )
21	19, 20	nemtbir 2453	. . 3 |- -. suc (/) ~~ (/)
22		ensymb 6834	. . 3 |- ( suc (/) ~~ (/) <-> (/) ~~ suc (/) )
23	21, 22	mtbi 671	. 2 |- -. (/) ~~ suc (/)
24		peano2 4627	. . . 4 |- ( k e. om -> suc k e. om )
25		vex 2763	. . . . 5 |- k e. _V
26	25	sucex 4531	. . . . 5 |- suc k e. _V
27	25, 26	phplem4 6911	. . . 4 |- ( ( k e. om /\ suc k e. om ) -> ( suc k ~~ suc suc k -> k ~~ suc k ) )
28	24, 27	mpdan 421	. . 3 |- ( k e. om -> ( suc k ~~ suc suc k -> k ~~ suc k ) )
29	28	con3d 632	. 2 |- ( k e. om -> ( -. k ~~ suc k -> -. suc k ~~ suc suc k ) )
30	4, 8, 12, 16, 23, 29	finds 4632	1 |- ( A e. om -> -. A ~~ suc A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   e. wcel 2164   =/= wne 2364   (/)c0 3446   class class class wbr 4029   suc csuc 4396   omcom 4622   ~~ cen 6792

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-er 6587  df-en 6795

This theorem is referenced by:  snnen2og  6915  1nen2  6917  php5dom  6919  php5fin  6938