Theorem php5 7015

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 4492	. . . 4 |- ( w = (/) -> suc w = suc (/) )
3	1, 2	breq12d 4095	. . 3 |- ( w = (/) -> ( w ~~ suc w <-> (/) ~~ suc (/) ) )
4	3	notbid 671	. 2 |- ( w = (/) -> ( -. w ~~ suc w <-> -. (/) ~~ suc (/) ) )
5		id 19	. . . 4 |- ( w = k -> w = k )
6		suceq 4492	. . . 4 |- ( w = k -> suc w = suc k )
7	5, 6	breq12d 4095	. . 3 |- ( w = k -> ( w ~~ suc w <-> k ~~ suc k ) )
8	7	notbid 671	. 2 |- ( w = k -> ( -. w ~~ suc w <-> -. k ~~ suc k ) )
9		id 19	. . . 4 |- ( w = suc k -> w = suc k )
10		suceq 4492	. . . 4 |- ( w = suc k -> suc w = suc suc k )
11	9, 10	breq12d 4095	. . 3 |- ( w = suc k -> ( w ~~ suc w <-> suc k ~~ suc suc k ) )
12	11	notbid 671	. 2 |- ( w = suc k -> ( -. w ~~ suc w <-> -. suc k ~~ suc suc k ) )
13		id 19	. . . 4 |- ( w = A -> w = A )
14		suceq 4492	. . . 4 |- ( w = A -> suc w = suc A )
15	13, 14	breq12d 4095	. . 3 |- ( w = A -> ( w ~~ suc w <-> A ~~ suc A ) )
16	15	notbid 671	. 2 |- ( w = A -> ( -. w ~~ suc w <-> -. A ~~ suc A ) )
17		peano1 4685	. . . . 5 |- (/) e. om
18		peano3 4687	. . . . 5 |- ( (/) e. om -> suc (/) =/= (/) )
19	17, 18	ax-mp 5	. . . 4 |- suc (/) =/= (/)
20		en0 6945	. . . 4 |- ( suc (/) ~~ (/) <-> suc (/) = (/) )
21	19, 20	nemtbir 2489	. . 3 |- -. suc (/) ~~ (/)
22		ensymb 6930	. . 3 |- ( suc (/) ~~ (/) <-> (/) ~~ suc (/) )
23	21, 22	mtbi 674	. 2 |- -. (/) ~~ suc (/)
24		peano2 4686	. . . 4 |- ( k e. om -> suc k e. om )
25		vex 2802	. . . . 5 |- k e. _V
26	25	sucex 4590	. . . . 5 |- suc k e. _V
27	25, 26	phplem4 7012	. . . 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 634	. 2 |- ( k e. om -> ( -. k ~~ suc k -> -. suc k ~~ suc suc k ) )
30	4, 8, 12, 16, 23, 29	finds 4691	1 |- ( A e. om -> -. A ~~ suc A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3491   class class class wbr 4082   suc csuc 4455   omcom 4681   ~~ cen 6883

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886

This theorem is referenced by:  snnen2og  7016  1nen2  7018  php5dom  7020  php5fin  7040