Theorem 0elnn 4651

Description: A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)

Assertion

Ref	Expression
0elnn	|- ( A e. om -> ( A = (/) \/ (/) e. A ) )

Proof of Theorem 0elnn

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

Step	Hyp	Ref	Expression
1		eqeq1 2200	. . 3 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2		eleq2 2257	. . 3 |- ( x = (/) -> ( (/) e. x <-> (/) e. (/) ) )
3	1, 2	orbi12d 794	. 2 |- ( x = (/) -> ( ( x = (/) \/ (/) e. x ) <-> ( (/) = (/) \/ (/) e. (/) ) ) )
4		eqeq1 2200	. . 3 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5		eleq2 2257	. . 3 |- ( x = y -> ( (/) e. x <-> (/) e. y ) )
6	4, 5	orbi12d 794	. 2 |- ( x = y -> ( ( x = (/) \/ (/) e. x ) <-> ( y = (/) \/ (/) e. y ) ) )
7		eqeq1 2200	. . 3 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8		eleq2 2257	. . 3 |- ( x = suc y -> ( (/) e. x <-> (/) e. suc y ) )
9	7, 8	orbi12d 794	. 2 |- ( x = suc y -> ( ( x = (/) \/ (/) e. x ) <-> ( suc y = (/) \/ (/) e. suc y ) ) )
10		eqeq1 2200	. . 3 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11		eleq2 2257	. . 3 |- ( x = A -> ( (/) e. x <-> (/) e. A ) )
12	10, 11	orbi12d 794	. 2 |- ( x = A -> ( ( x = (/) \/ (/) e. x ) <-> ( A = (/) \/ (/) e. A ) ) )
13		eqid 2193	. . 3 |- (/) = (/)
14	13	orci 732	. 2 |- ( (/) = (/) \/ (/) e. (/) )
15		0ex 4156	. . . . . . 7 |- (/) e. _V
16	15	sucid 4448	. . . . . 6 |- (/) e. suc (/)
17		suceq 4433	. . . . . 6 |- ( y = (/) -> suc y = suc (/) )
18	16, 17	eleqtrrid 2283	. . . . 5 |- ( y = (/) -> (/) e. suc y )
19	18	a1i 9	. . . 4 |- ( y e. om -> ( y = (/) -> (/) e. suc y ) )
20		sssucid 4446	. . . . . 6 |- y C_ suc y
21	20	a1i 9	. . . . 5 |- ( y e. om -> y C_ suc y )
22	21	sseld 3178	. . . 4 |- ( y e. om -> ( (/) e. y -> (/) e. suc y ) )
23	19, 22	jaod 718	. . 3 |- ( y e. om -> ( ( y = (/) \/ (/) e. y ) -> (/) e. suc y ) )
24		olc 712	. . 3 |- ( (/) e. suc y -> ( suc y = (/) \/ (/) e. suc y ) )
25	23, 24	syl6 33	. 2 |- ( y e. om -> ( ( y = (/) \/ (/) e. y ) -> ( suc y = (/) \/ (/) e. suc y ) ) )
26	3, 6, 9, 12, 14, 25	finds 4632	1 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2164   C_ wss 3153   (/)c0 3446   suc csuc 4396   omcom 4622

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623

This theorem is referenced by:  nn0eln0  4652  nnsucsssuc  6545  nntri3or  6546  nnm00  6583  ssfilem  6931  diffitest  6943  fiintim  6985  enumct  7174  nnnninfeq  7187  elni2  7374  enq0tr  7494  bj-charfunr  15302