Theorem nndceq0 4654

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 2203	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2	1	notbid 668	. . . 4 |- ( x = (/) -> ( -. x = (/) <-> -. (/) = (/) ) )
3	1, 2	orbi12d 794	. . 3 |- ( x = (/) -> ( ( x = (/) \/ -. x = (/) ) <-> ( (/) = (/) \/ -. (/) = (/) ) ) )
4		eqeq1 2203	. . . 4 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5	4	notbid 668	. . . 4 |- ( x = y -> ( -. x = (/) <-> -. y = (/) ) )
6	4, 5	orbi12d 794	. . 3 |- ( x = y -> ( ( x = (/) \/ -. x = (/) ) <-> ( y = (/) \/ -. y = (/) ) ) )
7		eqeq1 2203	. . . 4 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8	7	notbid 668	. . . 4 |- ( x = suc y -> ( -. x = (/) <-> -. suc y = (/) ) )
9	7, 8	orbi12d 794	. . 3 |- ( x = suc y -> ( ( x = (/) \/ -. x = (/) ) <-> ( suc y = (/) \/ -. suc y = (/) ) ) )
10		eqeq1 2203	. . . 4 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11	10	notbid 668	. . . 4 |- ( x = A -> ( -. x = (/) <-> -. A = (/) ) )
12	10, 11	orbi12d 794	. . 3 |- ( x = A -> ( ( x = (/) \/ -. x = (/) ) <-> ( A = (/) \/ -. A = (/) ) ) )
13		eqid 2196	. . . 4 |- (/) = (/)
14	13	orci 732	. . 3 |- ( (/) = (/) \/ -. (/) = (/) )
15		peano3 4632	. . . . . 6 |- ( y e. om -> suc y =/= (/) )
16	15	neneqd 2388	. . . . 5 |- ( y e. om -> -. suc y = (/) )
17	16	olcd 735	. . . 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 4636	. 2 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
20		df-dc 836	. 2 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
21	19, 20	sylibr 134	1 |- ( A e. om -> DECID A = (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   (/)c0 3450   suc csuc 4400   omcom 4626

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-iom 4627

This theorem is referenced by:  omp1eomlem  7160  ctmlemr  7174  nnnninfeq2  7195  nninfisol  7199  elni2  7381  indpi  7409  nnsf  15649  peano4nninf  15650