Theorem 0elnn 4717

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 2238	. . 3 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
2		eleq2 2295	. . 3 |- ( x = (/) -> ( (/) e. x <-> (/) e. (/) ) )
3	1, 2	orbi12d 800	. 2 |- ( x = (/) -> ( ( x = (/) \/ (/) e. x ) <-> ( (/) = (/) \/ (/) e. (/) ) ) )
4		eqeq1 2238	. . 3 |- ( x = y -> ( x = (/) <-> y = (/) ) )
5		eleq2 2295	. . 3 |- ( x = y -> ( (/) e. x <-> (/) e. y ) )
6	4, 5	orbi12d 800	. 2 |- ( x = y -> ( ( x = (/) \/ (/) e. x ) <-> ( y = (/) \/ (/) e. y ) ) )
7		eqeq1 2238	. . 3 |- ( x = suc y -> ( x = (/) <-> suc y = (/) ) )
8		eleq2 2295	. . 3 |- ( x = suc y -> ( (/) e. x <-> (/) e. suc y ) )
9	7, 8	orbi12d 800	. 2 |- ( x = suc y -> ( ( x = (/) \/ (/) e. x ) <-> ( suc y = (/) \/ (/) e. suc y ) ) )
10		eqeq1 2238	. . 3 |- ( x = A -> ( x = (/) <-> A = (/) ) )
11		eleq2 2295	. . 3 |- ( x = A -> ( (/) e. x <-> (/) e. A ) )
12	10, 11	orbi12d 800	. 2 |- ( x = A -> ( ( x = (/) \/ (/) e. x ) <-> ( A = (/) \/ (/) e. A ) ) )
13		eqid 2231	. . 3 |- (/) = (/)
14	13	orci 738	. 2 |- ( (/) = (/) \/ (/) e. (/) )
15		0ex 4216	. . . . . . 7 |- (/) e. _V
16	15	sucid 4514	. . . . . 6 |- (/) e. suc (/)
17		suceq 4499	. . . . . 6 |- ( y = (/) -> suc y = suc (/) )
18	16, 17	eleqtrrid 2321	. . . . 5 |- ( y = (/) -> (/) e. suc y )
19	18	a1i 9	. . . 4 |- ( y e. om -> ( y = (/) -> (/) e. suc y ) )
20		sssucid 4512	. . . . . 6 |- y C_ suc y
21	20	a1i 9	. . . . 5 |- ( y e. om -> y C_ suc y )
22	21	sseld 3226	. . . 4 |- ( y e. om -> ( (/) e. y -> (/) e. suc y ) )
23	19, 22	jaod 724	. . 3 |- ( y e. om -> ( ( y = (/) \/ (/) e. y ) -> (/) e. suc y ) )
24		olc 718	. . 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 4698	1 |- ( A e. om -> ( A = (/) \/ (/) e. A ) )

Syntax hints:   -> wi 4   \/ wo 715   = wceq 1397   e. wcel 2202   C_ wss 3200   (/)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-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  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:  nn0eln0  4718  nnsucsssuc  6659  nntri3or  6660  nnm00  6697  ssfilem  7061  ssfilemd  7063  diffitest  7075  fiintim  7122  enumct  7313  nnnninfeq  7326  elni2  7533  enq0tr  7653  bj-charfunr  16405