Theorem bj-nnelirr 15889

Description: A natural number does not belong to itself. Version of elirr 4589 for natural numbers, which does not require ax-setind 4585. (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 3464	. 2 ⊢ ¬ ∅ ∈ ∅
2		df-suc 4418	. . . . . 6 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
3	2	eleq2i 2272	. . . . 5 ⊢ (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4		elun 3314	. . . . . 6 ⊢ (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦 ∈ 𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5		bj-nntrans 15887	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ 𝑦 → suc 𝑦 ⊆ 𝑦))
6		sucssel 4471	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (suc 𝑦 ⊆ 𝑦 → 𝑦 ∈ 𝑦))
7	5, 6	syld 45	. . . . . . 7 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ 𝑦 → 𝑦 ∈ 𝑦))
8		vex 2775	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	8	sucid 4464	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
10		elsni 3651	. . . . . . . . 9 ⊢ (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
11	9, 10	eleqtrid 2294	. . . . . . . 8 ⊢ (suc 𝑦 ∈ {𝑦} → 𝑦 ∈ 𝑦)
12	11	a1i 9	. . . . . . 7 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦 ∈ 𝑦))
13	7, 12	jaod 719	. . . . . 6 ⊢ (𝑦 ∈ ω → ((suc 𝑦 ∈ 𝑦 ∨ suc 𝑦 ∈ {𝑦}) → 𝑦 ∈ 𝑦))
14	4, 13	biimtrid 152	. . . . 5 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) → 𝑦 ∈ 𝑦))
15	3, 14	biimtrid 152	. . . 4 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ suc 𝑦 → 𝑦 ∈ 𝑦))
16	15	con3d 632	. . 3 ⊢ (𝑦 ∈ ω → (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
17	16	rgen 2559	. 2 ⊢ ∀𝑦 ∈ ω (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18		ax-bdel 15757	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝑥
19	18	ax-bdn 15753	. . 3 ⊢ BOUNDED ¬ 𝑥 ∈ 𝑥
20		nfv 1551	. . 3 ⊢ Ⅎ𝑥 ¬ ∅ ∈ ∅
21		nfv 1551	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
22		nfv 1551	. . 3 ⊢ Ⅎ𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23		eleq1 2268	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
24		eleq2 2269	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
25	23, 24	bitrd 188	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑥 ↔ ∅ ∈ ∅))
26	25	notbid 669	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ 𝑥 ↔ ¬ ∅ ∈ ∅))
27	26	biimprd 158	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥 ∈ 𝑥))
28		elequ1 2180	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
29		elequ2 2181	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
30	28, 29	bitrd 188	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
31	30	notbid 669	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
32	31	biimpd 144	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑦))
33		eleq1 2268	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝑥 ↔ suc 𝑦 ∈ 𝑥))
34		eleq2 2269	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
35	33, 34	bitrd 188	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
36	35	notbid 669	. . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
37	36	biimprd 158	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥 ∈ 𝑥))
38		nfcv 2348	. . 3 ⊢ Ⅎ𝑥𝐴
39		nfv 1551	. . 3 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐴
40		eleq1 2268	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
41		eleq2 2269	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
42	40, 41	bitrd 188	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
43	42	notbid 669	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
44	43	biimpd 144	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴))
45	19, 20, 21, 22, 27, 32, 37, 38, 39, 44	bj-bdfindisg 15884	. 2 ⊢ ((¬ ∅ ∈ ∅ ∧ ∀𝑦 ∈ ω (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)) → (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴))
46	1, 17, 45	mp2an 426	1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 710   = wceq 1373   ∈ wcel 2176  ∀wral 2484   ∪ cun 3164   ⊆ wss 3166  ∅c0 3460  {csn 3633  suc csuc 4412  ωcom 4638

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4170  ax-pr 4253  ax-un 4480  ax-bd0 15749  ax-bdor 15752  ax-bdn 15753  ax-bdal 15754  ax-bdex 15755  ax-bdeq 15756  ax-bdel 15757  ax-bdsb 15758  ax-bdsep 15820  ax-infvn 15877

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639  df-bdc 15777  df-bj-ind 15863

This theorem is referenced by:  bj-nnen2lp  15890