Theorem 0elnn 4267

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 ∈ 𝜔 → (A = ∅ ∨ ∅ ∈ A))

Proof of Theorem 0elnn

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

Step	Hyp	Ref	Expression
1		eqeq1 2028	. . 3 ⊢ (x = ∅ → (x = ∅ ↔ ∅ = ∅))
2		eleq2 2083	. . 3 ⊢ (x = ∅ → (∅ ∈ x ↔ ∅ ∈ ∅))
3	1, 2	orbi12d 694	. 2 ⊢ (x = ∅ → ((x = ∅ ∨ ∅ ∈ x) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4		eqeq1 2028	. . 3 ⊢ (x = y → (x = ∅ ↔ y = ∅))
5		eleq2 2083	. . 3 ⊢ (x = y → (∅ ∈ x ↔ ∅ ∈ y))
6	4, 5	orbi12d 694	. 2 ⊢ (x = y → ((x = ∅ ∨ ∅ ∈ x) ↔ (y = ∅ ∨ ∅ ∈ y)))
7		eqeq1 2028	. . 3 ⊢ (x = suc y → (x = ∅ ↔ suc y = ∅))
8		eleq2 2083	. . 3 ⊢ (x = suc y → (∅ ∈ x ↔ ∅ ∈ suc y))
9	7, 8	orbi12d 694	. 2 ⊢ (x = suc y → ((x = ∅ ∨ ∅ ∈ x) ↔ (suc y = ∅ ∨ ∅ ∈ suc y)))
10		eqeq1 2028	. . 3 ⊢ (x = A → (x = ∅ ↔ A = ∅))
11		eleq2 2083	. . 3 ⊢ (x = A → (∅ ∈ x ↔ ∅ ∈ A))
12	10, 11	orbi12d 694	. 2 ⊢ (x = A → ((x = ∅ ∨ ∅ ∈ x) ↔ (A = ∅ ∨ ∅ ∈ A)))
13		eqid 2022	. . 3 ⊢ ∅ = ∅
14	13	orci 637	. 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅)
15		0ex 3858	. . . . . . 7 ⊢ ∅ ∈ V
16	15	sucid 4103	. . . . . 6 ⊢ ∅ ∈ suc ∅
17		suceq 4088	. . . . . 6 ⊢ (y = ∅ → suc y = suc ∅)
18	16, 17	syl5eleqr 2109	. . . . 5 ⊢ (y = ∅ → ∅ ∈ suc y)
19	18	a1i 9	. . . 4 ⊢ (y ∈ 𝜔 → (y = ∅ → ∅ ∈ suc y))
20		sssucid 4101	. . . . . 6 ⊢ y ⊆ suc y
21	20	a1i 9	. . . . 5 ⊢ (y ∈ 𝜔 → y ⊆ suc y)
22	21	sseld 2921	. . . 4 ⊢ (y ∈ 𝜔 → (∅ ∈ y → ∅ ∈ suc y))
23	19, 22	jaod 624	. . 3 ⊢ (y ∈ 𝜔 → ((y = ∅ ∨ ∅ ∈ y) → ∅ ∈ suc y))
24		olc 619	. . 3 ⊢ (∅ ∈ suc y → (suc y = ∅ ∨ ∅ ∈ suc y))
25	23, 24	syl6 29	. 2 ⊢ (y ∈ 𝜔 → ((y = ∅ ∨ ∅ ∈ y) → (suc y = ∅ ∨ ∅ ∈ suc y)))
26	3, 6, 9, 12, 14, 25	finds 4250	1 ⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∅ ∈ A))

Syntax hints:   → wi 4   ∨ wo 616   = wceq 1228   ∈ wcel 1374   ⊆ wss 2894  ∅c0 3201  suc csuc 4051  𝜔com 4240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-iinf 4238

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241

This theorem is referenced by:  nn0eln0  4268  nnsucsssuc  5986  nntri3or  5987  nnm00  6013  elni2  6174  enq0tr  6289