Theorem 0elnn 4283

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 2043	. . 3 ⊢ (x = ∅ → (x = ∅ ↔ ∅ = ∅))
2		eleq2 2098	. . 3 ⊢ (x = ∅ → (∅ ∈ x ↔ ∅ ∈ ∅))
3	1, 2	orbi12d 706	. 2 ⊢ (x = ∅ → ((x = ∅ ∨ ∅ ∈ x) ↔ (∅ = ∅ ∨ ∅ ∈ ∅)))
4		eqeq1 2043	. . 3 ⊢ (x = y → (x = ∅ ↔ y = ∅))
5		eleq2 2098	. . 3 ⊢ (x = y → (∅ ∈ x ↔ ∅ ∈ y))
6	4, 5	orbi12d 706	. 2 ⊢ (x = y → ((x = ∅ ∨ ∅ ∈ x) ↔ (y = ∅ ∨ ∅ ∈ y)))
7		eqeq1 2043	. . 3 ⊢ (x = suc y → (x = ∅ ↔ suc y = ∅))
8		eleq2 2098	. . 3 ⊢ (x = suc y → (∅ ∈ x ↔ ∅ ∈ suc y))
9	7, 8	orbi12d 706	. 2 ⊢ (x = suc y → ((x = ∅ ∨ ∅ ∈ x) ↔ (suc y = ∅ ∨ ∅ ∈ suc y)))
10		eqeq1 2043	. . 3 ⊢ (x = A → (x = ∅ ↔ A = ∅))
11		eleq2 2098	. . 3 ⊢ (x = A → (∅ ∈ x ↔ ∅ ∈ A))
12	10, 11	orbi12d 706	. 2 ⊢ (x = A → ((x = ∅ ∨ ∅ ∈ x) ↔ (A = ∅ ∨ ∅ ∈ A)))
13		eqid 2037	. . 3 ⊢ ∅ = ∅
14	13	orci 649	. 2 ⊢ (∅ = ∅ ∨ ∅ ∈ ∅)
15		0ex 3875	. . . . . . 7 ⊢ ∅ ∈ V
16	15	sucid 4120	. . . . . 6 ⊢ ∅ ∈ suc ∅
17		suceq 4105	. . . . . 6 ⊢ (y = ∅ → suc y = suc ∅)
18	16, 17	syl5eleqr 2124	. . . . 5 ⊢ (y = ∅ → ∅ ∈ suc y)
19	18	a1i 9	. . . 4 ⊢ (y ∈ 𝜔 → (y = ∅ → ∅ ∈ suc y))
20		sssucid 4118	. . . . . 6 ⊢ y ⊆ suc y
21	20	a1i 9	. . . . 5 ⊢ (y ∈ 𝜔 → y ⊆ suc y)
22	21	sseld 2938	. . . 4 ⊢ (y ∈ 𝜔 → (∅ ∈ y → ∅ ∈ suc y))
23	19, 22	jaod 636	. . 3 ⊢ (y ∈ 𝜔 → ((y = ∅ ∨ ∅ ∈ y) → ∅ ∈ suc y))
24		olc 631	. . 3 ⊢ (∅ ∈ suc y → (suc y = ∅ ∨ ∅ ∈ suc y))
25	23, 24	syl6 29	. 2 ⊢ (y ∈ 𝜔 → ((y = ∅ ∨ ∅ ∈ y) → (suc y = ∅ ∨ ∅ ∈ suc y)))
26	3, 6, 9, 12, 14, 25	finds 4266	1 ⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∅ ∈ A))

Syntax hints:   → wi 4   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  ∅c0 3218  suc csuc 4068  𝜔com 4256

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257

This theorem is referenced by:  nn0eln0  4284  nnsucsssuc  6010  nntri3or  6011  nnm00  6038  ssfiexmid  6254  elni2  6298  enq0tr  6417