Theorem bj-nnelirr 10748

Description: A natural number does not belong to itself. Version of elirr 4284 for natural numbers, which does not require ax-setind 4280. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nnelirr	⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)

Proof of Theorem bj-nnelirr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3255	. 2 ⊢ ¬ ∅ ∈ ∅
2		df-suc 4126	. . . . . 6 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
3	2	eleq2i 2145	. . . . 5 ⊢ (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4		elun 3113	. . . . . 6 ⊢ (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦 ∈ 𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5		bj-nntrans 10746	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ 𝑦 → suc 𝑦 ⊆ 𝑦))
6		sucssel 4179	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (suc 𝑦 ⊆ 𝑦 → 𝑦 ∈ 𝑦))
7	5, 6	syld 44	. . . . . . 7 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ 𝑦 → 𝑦 ∈ 𝑦))
8		vex 2604	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	8	sucid 4172	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
10		elsni 3416	. . . . . . . . 9 ⊢ (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
11	9, 10	syl5eleq 2167	. . . . . . . 8 ⊢ (suc 𝑦 ∈ {𝑦} → 𝑦 ∈ 𝑦)
12	11	a1i 9	. . . . . . 7 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦 ∈ 𝑦))
13	7, 12	jaod 669	. . . . . 6 ⊢ (𝑦 ∈ ω → ((suc 𝑦 ∈ 𝑦 ∨ suc 𝑦 ∈ {𝑦}) → 𝑦 ∈ 𝑦))
14	4, 13	syl5bi 150	. . . . 5 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) → 𝑦 ∈ 𝑦))
15	3, 14	syl5bi 150	. . . 4 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ suc 𝑦 → 𝑦 ∈ 𝑦))
16	15	con3d 593	. . 3 ⊢ (𝑦 ∈ ω → (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
17	16	rgen 2416	. 2 ⊢ ∀𝑦 ∈ ω (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18		ax-bdel 10612	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝑥
19	18	ax-bdn 10608	. . 3 ⊢ BOUNDED ¬ 𝑥 ∈ 𝑥
20		nfv 1461	. . 3 ⊢ Ⅎ𝑥 ¬ ∅ ∈ ∅
21		nfv 1461	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
22		nfv 1461	. . 3 ⊢ Ⅎ𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23		eleq1 2141	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
24		eleq2 2142	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
25	23, 24	bitrd 186	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑥 ↔ ∅ ∈ ∅))
26	25	notbid 624	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ 𝑥 ↔ ¬ ∅ ∈ ∅))
27	26	biimprd 156	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥 ∈ 𝑥))
28		elequ1 1640	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
29		elequ2 1641	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
30	28, 29	bitrd 186	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
31	30	notbid 624	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
32	31	biimpd 142	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑦))
33		eleq1 2141	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝑥 ↔ suc 𝑦 ∈ 𝑥))
34		eleq2 2142	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
35	33, 34	bitrd 186	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
36	35	notbid 624	. . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
37	36	biimprd 156	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥 ∈ 𝑥))
38		nfcv 2219	. . 3 ⊢ Ⅎ𝑥𝐴
39		nfv 1461	. . 3 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐴
40		eleq1 2141	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
41		eleq2 2142	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
42	40, 41	bitrd 186	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
43	42	notbid 624	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
44	43	biimpd 142	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴))
45	19, 20, 21, 22, 27, 32, 37, 38, 39, 44	bj-bdfindisg 10743	. 2 ⊢ ((¬ ∅ ∈ ∅ ∧ ∀𝑦 ∈ ω (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)) → (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴))
46	1, 17, 45	mp2an 416	1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 661   = wceq 1284   ∈ wcel 1433  ∀wral 2348   ∪ cun 2971   ⊆ wss 2973  ∅c0 3251  {csn 3398  suc csuc 4120  ωcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904  ax-pr 3964  ax-un 4188  ax-bd0 10604  ax-bdor 10607  ax-bdn 10608  ax-bdal 10609  ax-bdex 10610  ax-bdeq 10611  ax-bdel 10612  ax-bdsb 10613  ax-bdsep 10675  ax-infvn 10736

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332  df-bdc 10632  df-bj-ind 10722

This theorem is referenced by:  bj-nnen2lp  10749