Theorem bj-nnelirr 13988

Description: A natural number does not belong to itself. Version of elirr 4525 for natural numbers, which does not require ax-setind 4521. (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 3418	. 2 ⊢ ¬ ∅ ∈ ∅
2		df-suc 4356	. . . . . 6 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
3	2	eleq2i 2237	. . . . 5 ⊢ (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4		elun 3268	. . . . . 6 ⊢ (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦 ∈ 𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5		bj-nntrans 13986	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ 𝑦 → suc 𝑦 ⊆ 𝑦))
6		sucssel 4409	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (suc 𝑦 ⊆ 𝑦 → 𝑦 ∈ 𝑦))
7	5, 6	syld 45	. . . . . . 7 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ 𝑦 → 𝑦 ∈ 𝑦))
8		vex 2733	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	8	sucid 4402	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
10		elsni 3601	. . . . . . . . 9 ⊢ (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
11	9, 10	eleqtrid 2259	. . . . . . . 8 ⊢ (suc 𝑦 ∈ {𝑦} → 𝑦 ∈ 𝑦)
12	11	a1i 9	. . . . . . 7 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦 ∈ 𝑦))
13	7, 12	jaod 712	. . . . . 6 ⊢ (𝑦 ∈ ω → ((suc 𝑦 ∈ 𝑦 ∨ suc 𝑦 ∈ {𝑦}) → 𝑦 ∈ 𝑦))
14	4, 13	syl5bi 151	. . . . 5 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) → 𝑦 ∈ 𝑦))
15	3, 14	syl5bi 151	. . . 4 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ suc 𝑦 → 𝑦 ∈ 𝑦))
16	15	con3d 626	. . 3 ⊢ (𝑦 ∈ ω → (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
17	16	rgen 2523	. 2 ⊢ ∀𝑦 ∈ ω (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18		ax-bdel 13856	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝑥
19	18	ax-bdn 13852	. . 3 ⊢ BOUNDED ¬ 𝑥 ∈ 𝑥
20		nfv 1521	. . 3 ⊢ Ⅎ𝑥 ¬ ∅ ∈ ∅
21		nfv 1521	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
22		nfv 1521	. . 3 ⊢ Ⅎ𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23		eleq1 2233	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
24		eleq2 2234	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
25	23, 24	bitrd 187	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑥 ↔ ∅ ∈ ∅))
26	25	notbid 662	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ 𝑥 ↔ ¬ ∅ ∈ ∅))
27	26	biimprd 157	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥 ∈ 𝑥))
28		elequ1 2145	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
29		elequ2 2146	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
30	28, 29	bitrd 187	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
31	30	notbid 662	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
32	31	biimpd 143	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑦))
33		eleq1 2233	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝑥 ↔ suc 𝑦 ∈ 𝑥))
34		eleq2 2234	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
35	33, 34	bitrd 187	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
36	35	notbid 662	. . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
37	36	biimprd 157	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥 ∈ 𝑥))
38		nfcv 2312	. . 3 ⊢ Ⅎ𝑥𝐴
39		nfv 1521	. . 3 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐴
40		eleq1 2233	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
41		eleq2 2234	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
42	40, 41	bitrd 187	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
43	42	notbid 662	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
44	43	biimpd 143	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴))
45	19, 20, 21, 22, 27, 32, 37, 38, 39, 44	bj-bdfindisg 13983	. 2 ⊢ ((¬ ∅ ∈ ∅ ∧ ∀𝑦 ∈ ω (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)) → (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴))
46	1, 17, 45	mp2an 424	1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 703   = wceq 1348   ∈ wcel 2141  ∀wral 2448   ∪ cun 3119   ⊆ wss 3121  ∅c0 3414  {csn 3583  suc csuc 4350  ωcom 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdor 13851  ax-bdn 13852  ax-bdal 13853  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919  ax-infvn 13976

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962

This theorem is referenced by:  bj-nnen2lp  13989