Theorem nndceq0 4722

Description: A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)

Assertion

Ref	Expression
nndceq0	|- ( A e. om -> DECID A = (/) )

Proof of Theorem nndceq0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2238	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2	1	notbid 673	. . . 4 |- ( x = (/) -> ( -. x = (/) <-> -. (/) = (/) ) )
3	1, 2	orbi12d 801	. . 3 |- ( x = (/) -> ( ( x = (/) \/ -. x = (/) ) <-> ( (/) = (/) \/ -. (/) = (/) ) ) )
4		eqeq1 2238	. . . 4 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5	4	notbid 673	. . . 4 |- ( x = y -> ( -. x = (/) <-> -. y = (/) ) )
6	4, 5	orbi12d 801	. . 3 |- ( x = y -> ( ( x = (/) \/ -. x = (/) ) <-> ( y = (/) \/ -. y = (/) ) ) )
7		eqeq1 2238	. . . 4 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8	7	notbid 673	. . . 4 |- ( x = suc y -> ( -. x = (/) <-> -. suc y = (/) ) )
9	7, 8	orbi12d 801	. . 3 |- ( x = suc y -> ( ( x = (/) \/ -. x = (/) ) <-> ( suc y = (/) \/ -. suc y = (/) ) ) )
10		eqeq1 2238	. . . 4 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11	10	notbid 673	. . . 4 |- ( x = A -> ( -. x = (/) <-> -. A = (/) ) )
12	10, 11	orbi12d 801	. . 3 |- ( x = A -> ( ( x = (/) \/ -. x = (/) ) <-> ( A = (/) \/ -. A = (/) ) ) )
13		eqid 2231	. . . 4 |- (/) = (/)
14	13	orci 739	. . 3 |- ( (/) = (/) \/ -. (/) = (/) )
15		peano3 4700	. . . . . 6 |- ( y e. om -> suc y =/= (/) )
16	15	neneqd 2424	. . . . 5 |- ( y e. om -> -. suc y = (/) )
17	16	olcd 742	. . . 4 |- ( y e. om -> ( suc y = (/) \/ -. suc y = (/) ) )
18	17	a1d 22	. . 3 |- ( y e. om -> ( ( y = (/) \/ -. y = (/) ) -> ( suc y = (/) \/ -. suc y = (/) ) ) )
19	3, 6, 9, 12, 14, 18	finds 4704	. 2 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
20		df-dc 843	. 2 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
21	19, 20	sylibr 134	1 |- ( A e. om -> DECID A = (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2202   (/)c0 3496   suc csuc 4468   omcom 4694

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-suc 4474  df-iom 4695

This theorem is referenced by:  omp1eomlem  7353  ctmlemr  7367  nnnninfeq2  7388  nninfisol  7392  elni2  7594  indpi  7622  nnsf  16731  peano4nninf  16732