Theorem php5 7039

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 4497	. . . 4 |- ( w = (/) -> suc w = suc (/) )
3	1, 2	breq12d 4099	. . 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 4497	. . . 4 |- ( w = k -> suc w = suc k )
7	5, 6	breq12d 4099	. . 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 4497	. . . 4 |- ( w = suc k -> suc w = suc suc k )
11	9, 10	breq12d 4099	. . 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 4497	. . . 4 |- ( w = A -> suc w = suc A )
15	13, 14	breq12d 4099	. . 3 |- ( w = A -> ( w ~~ suc w <-> A ~~ suc A ) )
16	15	notbid 671	. 2 |- ( w = A -> ( -. w ~~ suc w <-> -. A ~~ suc A ) )
17		peano1 4690	. . . . 5 |- (/) e. om
18		peano3 4692	. . . . 5 |- ( (/) e. om -> suc (/) =/= (/) )
19	17, 18	ax-mp 5	. . . 4 |- suc (/) =/= (/)
20		en0 6964	. . . 4 |- ( suc (/) ~~ (/) <-> suc (/) = (/) )
21	19, 20	nemtbir 2489	. . 3 |- -. suc (/) ~~ (/)
22		ensymb 6949	. . 3 |- ( suc (/) ~~ (/) <-> (/) ~~ suc (/) )
23	21, 22	mtbi 674	. 2 |- -. (/) ~~ suc (/)
24		peano2 4691	. . . 4 |- ( k e. om -> suc k e. om )
25		vex 2803	. . . . 5 |- k e. _V
26	25	sucex 4595	. . . . 5 |- suc k e. _V
27	25, 26	phplem4 7036	. . . 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 4696	1 |- ( A e. om -> -. A ~~ suc A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   e. wcel 2200   =/= wne 2400   (/)c0 3492   class class class wbr 4086   suc csuc 4460   omcom 4686   ~~ cen 6902

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 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-er 6697  df-en 6905

This theorem is referenced by:  snnen2og  7040  1nen2  7042  php5dom  7044  php5fin  7064