Theorem 0elnn 4470

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 2106	. . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2		eleq2 2163	. . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
3	1, 2	orbi12d 748	. 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4		eqeq1 2106	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5		eleq2 2163	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
6	4, 5	orbi12d 748	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7		eqeq1 2106	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8		eleq2 2163	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
9	7, 8	orbi12d 748	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10		eqeq1 2106	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11		eleq2 2163	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
12	10, 11	orbi12d 748	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13		eqid 2100	. . 3 ⊢ ∅ = ∅
14	13	orci 691	. 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅)
15		0ex 3995	. . . . . . 7 ⊢ ∅ ∈ V
16	15	sucid 4277	. . . . . 6 ⊢ ∅ ∈ suc ∅
17		suceq 4262	. . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
18	16, 17	syl5eleqr 2189	. . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦)
19	18	a1i 9	. . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20		sssucid 4275	. . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦
21	20	a1i 9	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
22	21	sseld 3046	. . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
23	19, 22	jaod 678	. . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24		olc 673	. . 3 ⊢ (∅ ∈ suc 𝑦 → (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦))
25	23, 24	syl6 33	. 2 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
26	3, 6, 9, 12, 14, 25	finds 4452	1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Syntax hints:   → wi 4   ∨ wo 670   = wceq 1299   ∈ wcel 1448   ⊆ wss 3021  ∅c0 3310  suc csuc 4225  ωcom 4442

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-iom 4443

This theorem is referenced by:  nn0eln0  4471  nnsucsssuc  6318  nntri3or  6319  nnm00  6355  ssfilem  6698  diffitest  6710  fiintim  6746  enumct  6914  elni2  7023  enq0tr  7143  nninfalllemn  12786