Theorem 0elnn 4766

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 2245	. . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2		eleq2 2302	. . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
3	1, 2	orbi12d 805	. 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4		eqeq1 2245	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5		eleq2 2302	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
6	4, 5	orbi12d 805	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7		eqeq1 2245	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8		eleq2 2302	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
9	7, 8	orbi12d 805	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10		eqeq1 2245	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11		eleq2 2302	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
12	10, 11	orbi12d 805	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13		eqid 2238	. . 3 ⊢ ∅ = ∅
14	13	orci 743	. 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅)
15		0ex 4260	. . . . . . 7 ⊢ ∅ ∈ V
16	15	sucid 4562	. . . . . 6 ⊢ ∅ ∈ suc ∅
17		suceq 4547	. . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
18	16, 17	eleqtrrid 2328	. . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦)
19	18	a1i 9	. . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20		sssucid 4560	. . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦
21	20	a1i 9	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
22	21	sseld 3247	. . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
23	19, 22	jaod 729	. . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24		olc 723	. . 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 4747	1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∨ wo 720   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  ∅c0 3520  suc csuc 4510  ωcom 4737

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-uni 3936  df-int 3971  df-suc 4516  df-iom 4738

This theorem is used by:  nn0eln0  4767  nnsucsssuc  6765  nntri3or  6766  nnm00  6803  ssfilem  7177  ssfilemd  7179  diffitest  7191  fiintim  7238  enumct  7455  nnnninfeq  7468  elni2  7681  enq0tr  7801  bj-charfunr  16836