Theorem nndceq0 4716

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 800	. . 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 800	. . 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 800	. . 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 800	. . 3 |- ( x = A -> ( ( x = (/) \/ -. x = (/) ) <-> ( A = (/) \/ -. A = (/) ) ) )
13		eqid 2231	. . . 4 |- (/) = (/)
14	13	orci 738	. . 3 |- ( (/) = (/) \/ -. (/) = (/) )
15		peano3 4694	. . . . . 6 |- ( y e. om -> suc y =/= (/) )
16	15	neneqd 2423	. . . . 5 |- ( y e. om -> -. suc y = (/) )
17	16	olcd 741	. . . 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 4698	. 2 |- ( A e. om -> ( A = (/) \/ -. A = (/) ) )
20		df-dc 842	. 2 |- (DECID A = (/) <-> ( A = (/) \/ -. A = (/) ) )
21	19, 20	sylibr 134	1 |- ( A e. om -> DECID A = (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   (/)c0 3494   suc csuc 4462   omcom 4688

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-int 3929  df-suc 4468  df-iom 4689

This theorem is referenced by:  omp1eomlem  7292  ctmlemr  7306  nnnninfeq2  7327  nninfisol  7331  elni2  7533  indpi  7561  nnsf  16607  peano4nninf  16608