Theorem 0elnn 4358

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 2087	. . 3 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2		eleq2 2142	. . 3 |- ( x = (/) -> ( (/) e. x <-> (/) e. (/) ) )
3	1, 2	orbi12d 739	. 2 |- ( x = (/) -> ( ( x = (/) \/ (/) e. x ) <-> ( (/) = (/) \/ (/) e. (/) ) ) )
4		eqeq1 2087	. . 3 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5		eleq2 2142	. . 3 |- ( x = y -> ( (/) e. x <-> (/) e. y ) )
6	4, 5	orbi12d 739	. 2 |- ( x = y -> ( ( x = (/) \/ (/) e. x ) <-> ( y = (/) \/ (/) e. y ) ) )
7		eqeq1 2087	. . 3 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8		eleq2 2142	. . 3 |- ( x = suc y -> ( (/) e. x <-> (/) e. suc y ) )
9	7, 8	orbi12d 739	. 2 |- ( x = suc y -> ( ( x = (/) \/ (/) e. x ) <-> ( suc y = (/) \/ (/) e. suc y ) ) )
10		eqeq1 2087	. . 3 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11		eleq2 2142	. . 3 |- ( x = A -> ( (/) e. x <-> (/) e. A ) )
12	10, 11	orbi12d 739	. 2 |- ( x = A -> ( ( x = (/) \/ (/) e. x ) <-> ( A = (/) \/ (/) e. A ) ) )
13		eqid 2081	. . 3 |- (/) = (/)
14	13	orci 682	. 2 |- ( (/) = (/) \/ (/) e. (/) )
15		0ex 3905	. . . . . . 7 |- (/) e. _V
16	15	sucid 4172	. . . . . 6 |- (/) e. suc (/)
17		suceq 4157	. . . . . 6 |- ( y = (/) -> suc y = suc (/) )
18	16, 17	syl5eleqr 2168	. . . . 5 |- ( y = (/) -> (/) e. suc y )
19	18	a1i 9	. . . 4 |- ( y e. om -> ( y = (/) -> (/) e. suc y ) )
20		sssucid 4170	. . . . . 6 |- y C_ suc y
21	20	a1i 9	. . . . 5 |- ( y e. om -> y C_ suc y )
22	21	sseld 2998	. . . 4 |- ( y e. om -> ( (/) e. y -> (/) e. suc y ) )
23	19, 22	jaod 669	. . 3 |- ( y e. om -> ( ( y = (/) \/ (/) e. y ) -> (/) e. suc y ) )
24		olc 664	. . 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 4341	1 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )

Syntax hints:   -> wi 4   \/ wo 661   = wceq 1284   e. wcel 1433   C_ wss 2973   (/)c0 3251   suc csuc 4120   omcom 4331

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332

This theorem is referenced by:  nn0eln0  4359  nnsucsssuc  6094  nntri3or  6095  nnm00  6125  ssfilem  6360  diffitest  6371  elni2  6504  enq0tr  6624