Theorem 0elnn 4675

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 2213	. . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2		eleq2 2270	. . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
3	1, 2	orbi12d 795	. 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4		eqeq1 2213	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5		eleq2 2270	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
6	4, 5	orbi12d 795	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7		eqeq1 2213	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8		eleq2 2270	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
9	7, 8	orbi12d 795	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10		eqeq1 2213	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11		eleq2 2270	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
12	10, 11	orbi12d 795	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13		eqid 2206	. . 3 ⊢ ∅ = ∅
14	13	orci 733	. 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅)
15		0ex 4179	. . . . . . 7 ⊢ ∅ ∈ V
16	15	sucid 4472	. . . . . 6 ⊢ ∅ ∈ suc ∅
17		suceq 4457	. . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
18	16, 17	eleqtrrid 2296	. . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦)
19	18	a1i 9	. . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20		sssucid 4470	. . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦
21	20	a1i 9	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
22	21	sseld 3196	. . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
23	19, 22	jaod 719	. . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24		olc 713	. . 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 4656	1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Syntax hints:   → wi 4   ∨ wo 710   = wceq 1373   ∈ wcel 2177   ⊆ wss 3170  ∅c0 3464  suc csuc 4420  ωcom 4646

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3857  df-int 3892  df-suc 4426  df-iom 4647

This theorem is referenced by:  nn0eln0  4676  nnsucsssuc  6591  nntri3or  6592  nnm00  6629  ssfilem  6987  diffitest  6999  fiintim  7043  enumct  7232  nnnninfeq  7245  elni2  7447  enq0tr  7567  bj-charfunr  15884