Theorem nndceq0 4431

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 2094	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2	1	notbid 627	. . . 4 |- ( x = (/) -> ( -. x = (/) <-> -. (/) = (/) ) )
3	1, 2	orbi12d 742	. . 3 |- ( x = (/) -> ( ( x = (/) \/ -. x = (/) ) <-> ( (/) = (/) \/ -. (/) = (/) ) ) )
4		eqeq1 2094	. . . 4 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5	4	notbid 627	. . . 4 |- ( x = y -> ( -. x = (/) <-> -. y = (/) ) )
6	4, 5	orbi12d 742	. . 3 |- ( x = y -> ( ( x = (/) \/ -. x = (/) ) <-> ( y = (/) \/ -. y = (/) ) ) )
7		eqeq1 2094	. . . 4 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8	7	notbid 627	. . . 4 |- ( x = suc y -> ( -. x = (/) <-> -. suc y = (/) ) )
9	7, 8	orbi12d 742	. . 3 |- ( x = suc y -> ( ( x = (/) \/ -. x = (/) ) <-> ( suc y = (/) \/ -. suc y = (/) ) ) )
10		eqeq1 2094	. . . 4 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11	10	notbid 627	. . . 4 |- ( x = A -> ( -. x = (/) <-> -. A = (/) ) )
12	10, 11	orbi12d 742	. . 3 |- ( x = A -> ( ( x = (/) \/ -. x = (/) ) <-> ( A = (/) \/ -. A = (/) ) ) )
13		eqid 2088	. . . 4 |- (/) = (/)
14	13	orci 685	. . 3 |- ( (/) = (/) \/ -. (/) = (/) )
15		peano3 4411	. . . . . 6 |- ( y e. om -> suc y =/= (/) )
16	15	neneqd 2276	. . . . 5 |- ( y e. om -> -. suc y = (/) )
17	16	olcd 688	. . . 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 4415	. 2 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
20		df-dc 781	. 2 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
21	19, 20	sylibr 132	1 |- ( A e. om -> DECID A = (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 664  DECID wdc 780   = wceq 1289   e. wcel 1438   (/)c0 3286   suc csuc 4192   omcom 4405

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-int 3689  df-suc 4198  df-iom 4406

This theorem is referenced by:  elni2  6871  indpi  6899  nnsf  11850  peano4nninf  11851