Theorem 0elnn 4746

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 2241	. . 3 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2		eleq2 2298	. . 3 |- ( x = (/) -> ( (/) e. x <-> (/) e. (/) ) )
3	1, 2	orbi12d 801	. 2 |- ( x = (/) -> ( ( x = (/) \/ (/) e. x ) <-> ( (/) = (/) \/ (/) e. (/) ) ) )
4		eqeq1 2241	. . 3 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5		eleq2 2298	. . 3 |- ( x = y -> ( (/) e. x <-> (/) e. y ) )
6	4, 5	orbi12d 801	. 2 |- ( x = y -> ( ( x = (/) \/ (/) e. x ) <-> ( y = (/) \/ (/) e. y ) ) )
7		eqeq1 2241	. . 3 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8		eleq2 2298	. . 3 |- ( x = suc y -> ( (/) e. x <-> (/) e. suc y ) )
9	7, 8	orbi12d 801	. 2 |- ( x = suc y -> ( ( x = (/) \/ (/) e. x ) <-> ( suc y = (/) \/ (/) e. suc y ) ) )
10		eqeq1 2241	. . 3 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11		eleq2 2298	. . 3 |- ( x = A -> ( (/) e. x <-> (/) e. A ) )
12	10, 11	orbi12d 801	. 2 |- ( x = A -> ( ( x = (/) \/ (/) e. x ) <-> ( A = (/) \/ (/) e. A ) ) )
13		eqid 2234	. . 3 |- (/) = (/)
14	13	orci 739	. 2 |- ( (/) = (/) \/ (/) e. (/) )
15		0ex 4242	. . . . . . 7 |- (/) e. _V
16	15	sucid 4543	. . . . . 6 |- (/) e. suc (/)
17		suceq 4528	. . . . . 6 |- ( y = (/) -> suc y = suc (/) )
18	16, 17	eleqtrrid 2324	. . . . 5 |- ( y = (/) -> (/) e. suc y )
19	18	a1i 9	. . . 4 |- ( y e. om -> ( y = (/) -> (/) e. suc y ) )
20		sssucid 4541	. . . . . 6 |- y C_ suc y
21	20	a1i 9	. . . . 5 |- ( y e. om -> y C_ suc y )
22	21	sseld 3241	. . . 4 |- ( y e. om -> ( (/) e. y -> (/) e. suc y ) )
23	19, 22	jaod 725	. . 3 |- ( y e. om -> ( ( y = (/) \/ (/) e. y ) -> (/) e. suc y ) )
24		olc 719	. . 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 4727	1 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2205   C_ wss 3214   (/)c0 3512   suc csuc 4491   omcom 4717

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-suc 4497  df-iom 4718

This theorem is referenced by:  nn0eln0  4747  nnsucsssuc  6738  nntri3or  6739  nnm00  6776  ssfilem  7143  ssfilemd  7145  diffitest  7157  fiintim  7204  enumct  7419  nnnninfeq  7432  elni2  7645  enq0tr  7765  bj-charfunr  16706