Theorem php5 6321

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 4139	. . . 4 |- ( w = (/) -> suc w = suc (/) )
3	1, 2	breq12d 3777	. . 3 |- ( w = (/) -> ( w ~~ suc w <-> (/) ~~ suc (/) ) )
4	3	notbid 592	. 2 |- ( w = (/) -> ( -. w ~~ suc w <-> -. (/) ~~ suc (/) ) )
5		id 19	. . . 4 |- ( w = k -> w = k )
6		suceq 4139	. . . 4 |- ( w = k -> suc w = suc k )
7	5, 6	breq12d 3777	. . 3 |- ( w = k -> ( w ~~ suc w <-> k ~~ suc k ) )
8	7	notbid 592	. 2 |- ( w = k -> ( -. w ~~ suc w <-> -. k ~~ suc k ) )
9		id 19	. . . 4 |- ( w = suc k -> w = suc k )
10		suceq 4139	. . . 4 |- ( w = suc k -> suc w = suc suc k )
11	9, 10	breq12d 3777	. . 3 |- ( w = suc k -> ( w ~~ suc w <-> suc k ~~ suc suc k ) )
12	11	notbid 592	. 2 |- ( w = suc k -> ( -. w ~~ suc w <-> -. suc k ~~ suc suc k ) )
13		id 19	. . . 4 |- ( w = A -> w = A )
14		suceq 4139	. . . 4 |- ( w = A -> suc w = suc A )
15	13, 14	breq12d 3777	. . 3 |- ( w = A -> ( w ~~ suc w <-> A ~~ suc A ) )
16	15	notbid 592	. 2 |- ( w = A -> ( -. w ~~ suc w <-> -. A ~~ suc A ) )
17		peano1 4317	. . . . 5 |- (/) e. om
18		peano3 4319	. . . . 5 |- ( (/) e. om -> suc (/) =/= (/) )
19	17, 18	ax-mp 7	. . . 4 |- suc (/) =/= (/)
20		en0 6275	. . . 4 |- ( suc (/) ~~ (/) <-> suc (/) = (/) )
21	19, 20	nemtbir 2294	. . 3 |- -. suc (/) ~~ (/)
22		ensymb 6260	. . 3 |- ( suc (/) ~~ (/) <-> (/) ~~ suc (/) )
23	21, 22	mtbi 595	. 2 |- -. (/) ~~ suc (/)
24		peano2 4318	. . . 4 |- ( k e. om -> suc k e. om )
25		vex 2560	. . . . 5 |- k e. _V
26	25	sucex 4225	. . . . 5 |- suc k e. _V
27	25, 26	phplem4 6318	. . . 4 |- ( ( k e. om /\ suc k e. om ) -> ( suc k ~~ suc suc k -> k ~~ suc k ) )
28	24, 27	mpdan 398	. . 3 |- ( k e. om -> ( suc k ~~ suc suc k -> k ~~ suc k ) )
29	28	con3d 561	. 2 |- ( k e. om -> ( -. k ~~ suc k -> -. suc k ~~ suc suc k ) )
30	4, 8, 12, 16, 23, 29	finds 4323	1 |- ( A e. om -> -. A ~~ suc A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1243   e. wcel 1393   =/= wne 2204   (/)c0 3224   class class class wbr 3764   suc csuc 4102   omcom 4313   ~~ cen 6219

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-er 6106  df-en 6222

This theorem is referenced by:  snnen2og  6322  php5dom  6325  php5fin  6339