Theorem nndceq0 4760

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 2245	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2	1	notbid 677	. . . 4 |- ( x = (/) -> ( -. x = (/) <-> -. (/) = (/) ) )
3	1, 2	orbi12d 805	. . 3 |- ( x = (/) -> ( ( x = (/) \/ -. x = (/) ) <-> ( (/) = (/) \/ -. (/) = (/) ) ) )
4		eqeq1 2245	. . . 4 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5	4	notbid 677	. . . 4 |- ( x = y -> ( -. x = (/) <-> -. y = (/) ) )
6	4, 5	orbi12d 805	. . 3 |- ( x = y -> ( ( x = (/) \/ -. x = (/) ) <-> ( y = (/) \/ -. y = (/) ) ) )
7		eqeq1 2245	. . . 4 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8	7	notbid 677	. . . 4 |- ( x = suc y -> ( -. x = (/) <-> -. suc y = (/) ) )
9	7, 8	orbi12d 805	. . 3 |- ( x = suc y -> ( ( x = (/) \/ -. x = (/) ) <-> ( suc y = (/) \/ -. suc y = (/) ) ) )
10		eqeq1 2245	. . . 4 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11	10	notbid 677	. . . 4 |- ( x = A -> ( -. x = (/) <-> -. A = (/) ) )
12	10, 11	orbi12d 805	. . 3 |- ( x = A -> ( ( x = (/) \/ -. x = (/) ) <-> ( A = (/) \/ -. A = (/) ) ) )
13		eqid 2238	. . . 4 |- (/) = (/)
14	13	orci 743	. . 3 |- ( (/) = (/) \/ -. (/) = (/) )
15		peano3 4738	. . . . . 6 |- ( y e. om -> suc y =/= (/) )
16	15	neneqd 2441	. . . . 5 |- ( y e. om -> -. suc y = (/) )
17	16	olcd 746	. . . 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 4742	. 2 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
20		df-dc 847	. 2 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
21	19, 20	sylibr 134	1 |- ( A e. om -> DECID A = (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   (/)c0 3520   suc csuc 4505   omcom 4732

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733

This theorem is referenced by:  omp1eomlem  7424  ctmlemr  7438  nnnninfeq2  7459  nninfisol  7463  elni2  7671  indpi  7699  nnsf  16953  peano4nninf  16954