Theorem 0elnn 4655

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 2203	. . 3 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2		eleq2 2260	. . 3 |- ( x = (/) -> ( (/) e. x <-> (/) e. (/) ) )
3	1, 2	orbi12d 794	. 2 |- ( x = (/) -> ( ( x = (/) \/ (/) e. x ) <-> ( (/) = (/) \/ (/) e. (/) ) ) )
4		eqeq1 2203	. . 3 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5		eleq2 2260	. . 3 |- ( x = y -> ( (/) e. x <-> (/) e. y ) )
6	4, 5	orbi12d 794	. 2 |- ( x = y -> ( ( x = (/) \/ (/) e. x ) <-> ( y = (/) \/ (/) e. y ) ) )
7		eqeq1 2203	. . 3 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8		eleq2 2260	. . 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 2203	. . 3 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11		eleq2 2260	. . 3 |- ( x = A -> ( (/) e. x <-> (/) e. A ) )
12	10, 11	orbi12d 794	. 2 |- ( x = A -> ( ( x = (/) \/ (/) e. x ) <-> ( A = (/) \/ (/) e. A ) ) )
13		eqid 2196	. . 3 |- (/) = (/)
14	13	orci 732	. 2 |- ( (/) = (/) \/ (/) e. (/) )
15		0ex 4160	. . . . . . 7 |- (/) e. _V
16	15	sucid 4452	. . . . . 6 |- (/) e. suc (/)
17		suceq 4437	. . . . . 6 |- ( y = (/) -> suc y = suc (/) )
18	16, 17	eleqtrrid 2286	. . . . 5 |- ( y = (/) -> (/) e. suc y )
19	18	a1i 9	. . . 4 |- ( y e. om -> ( y = (/) -> (/) e. suc y ) )
20		sssucid 4450	. . . . . 6 |- y C_ suc y
21	20	a1i 9	. . . . 5 |- ( y e. om -> y C_ suc y )
22	21	sseld 3182	. . . 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 4636	1 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2167   C_ wss 3157   (/)c0 3450   suc csuc 4400   omcom 4626

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-iom 4627

This theorem is referenced by:  nn0eln0  4656  nnsucsssuc  6550  nntri3or  6551  nnm00  6588  ssfilem  6936  diffitest  6948  fiintim  6992  enumct  7181  nnnninfeq  7194  elni2  7381  enq0tr  7501  bj-charfunr  15456