Theorem 0elnn 4620

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 2184	. . 3 ⊢ (𝑥 = ∅ → (𝑥 = ∅ ↔ ∅ = ∅))
2		eleq2 2241	. . 3 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
3	1, 2	orbi12d 793	. 2 ⊢ (𝑥 = ∅ → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4		eqeq1 2184	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5		eleq2 2241	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
6	4, 5	orbi12d 793	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝑦 = ∅ ∨ ∅ ∈ 𝑦)))
7		eqeq1 2184	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 = ∅ ↔ suc 𝑦 = ∅))
8		eleq2 2241	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
9	7, 8	orbi12d 793	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (suc 𝑦 = ∅ ∨ ∅ ∈ suc 𝑦)))
10		eqeq1 2184	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
11		eleq2 2241	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐴))
12	10, 11	orbi12d 793	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)))
13		eqid 2177	. . 3 ⊢ ∅ = ∅
14	13	orci 731	. 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅)
15		0ex 4132	. . . . . . 7 ⊢ ∅ ∈ V
16	15	sucid 4419	. . . . . 6 ⊢ ∅ ∈ suc ∅
17		suceq 4404	. . . . . 6 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
18	16, 17	eleqtrrid 2267	. . . . 5 ⊢ (𝑦 = ∅ → ∅ ∈ suc 𝑦)
19	18	a1i 9	. . . 4 ⊢ (𝑦 ∈ ω → (𝑦 = ∅ → ∅ ∈ suc 𝑦))
20		sssucid 4417	. . . . . 6 ⊢ 𝑦 ⊆ suc 𝑦
21	20	a1i 9	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ⊆ suc 𝑦)
22	21	sseld 3156	. . . 4 ⊢ (𝑦 ∈ ω → (∅ ∈ 𝑦 → ∅ ∈ suc 𝑦))
23	19, 22	jaod 717	. . 3 ⊢ (𝑦 ∈ ω → ((𝑦 = ∅ ∨ ∅ ∈ 𝑦) → ∅ ∈ suc 𝑦))
24		olc 711	. . 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 4601	1 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Syntax hints:   → wi 4   ∨ wo 708   = wceq 1353   ∈ wcel 2148   ⊆ wss 3131  ∅c0 3424  suc csuc 4367  ωcom 4591

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592

This theorem is referenced by:  nn0eln0  4621  nnsucsssuc  6495  nntri3or  6496  nnm00  6533  ssfilem  6877  diffitest  6889  fiintim  6930  enumct  7116  nnnninfeq  7128  elni2  7315  enq0tr  7435  bj-charfunr  14647