Theorem 0elnn 4710

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 2236	. . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2		eleq2 2293	. . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
3	1, 2	orbi12d 798	. 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4		eqeq1 2236	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5		eleq2 2293	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
6	4, 5	orbi12d 798	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7		eqeq1 2236	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8		eleq2 2293	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
9	7, 8	orbi12d 798	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10		eqeq1 2236	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11		eleq2 2293	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
12	10, 11	orbi12d 798	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13		eqid 2229	. . 3 ⊢ ∅ = ∅
14	13	orci 736	. 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅)
15		0ex 4210	. . . . . . 7 ⊢ ∅ ∈ V
16	15	sucid 4507	. . . . . 6 ⊢ ∅ ∈ suc ∅
17		suceq 4492	. . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
18	16, 17	eleqtrrid 2319	. . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦)
19	18	a1i 9	. . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20		sssucid 4505	. . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦
21	20	a1i 9	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
22	21	sseld 3223	. . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
23	19, 22	jaod 722	. . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24		olc 716	. . 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 4691	1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Syntax hints:   → wi 4   ∨ wo 713   = wceq 1395   ∈ wcel 2200   ⊆ wss 3197  ∅c0 3491  suc csuc 4455  ωcom 4681

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923  df-suc 4461  df-iom 4682

This theorem is referenced by:  nn0eln0  4711  nnsucsssuc  6636  nntri3or  6637  nnm00  6674  ssfilem  7033  diffitest  7045  fiintim  7089  enumct  7278  nnnninfeq  7291  elni2  7497  enq0tr  7617  bj-charfunr  16131