Theorem bj-nnelirr 10078

Description: A natural number does not belong to itself. Version of elirr 4266 for natural numbers, which does not require ax-setind 4262. (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 3228	. 2 ⊢ ¬ ∅ ∈ ∅
2		df-suc 4108	. . . . . 6 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
3	2	eleq2i 2104	. . . . 5 ⊢ (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4		elun 3084	. . . . . 6 ⊢ (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦 ∈ 𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5		bj-nntrans 10076	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ 𝑦 → suc 𝑦 ⊆ 𝑦))
6		sucssel 4161	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (suc 𝑦 ⊆ 𝑦 → 𝑦 ∈ 𝑦))
7	5, 6	syld 40	. . . . . . 7 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ 𝑦 → 𝑦 ∈ 𝑦))
8		vex 2560	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	8	sucid 4154	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
10		elsni 3393	. . . . . . . . 9 ⊢ (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
11	9, 10	syl5eleq 2126	. . . . . . . 8 ⊢ (suc 𝑦 ∈ {𝑦} → 𝑦 ∈ 𝑦)
12	11	a1i 9	. . . . . . 7 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦 ∈ 𝑦))
13	7, 12	jaod 637	. . . . . 6 ⊢ (𝑦 ∈ ω → ((suc 𝑦 ∈ 𝑦 ∨ suc 𝑦 ∈ {𝑦}) → 𝑦 ∈ 𝑦))
14	4, 13	syl5bi 141	. . . . 5 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) → 𝑦 ∈ 𝑦))
15	3, 14	syl5bi 141	. . . 4 ⊢ (𝑦 ∈ ω → (suc 𝑦 ∈ suc 𝑦 → 𝑦 ∈ 𝑦))
16	15	con3d 561	. . 3 ⊢ (𝑦 ∈ ω → (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
17	16	rgen 2374	. 2 ⊢ ∀𝑦 ∈ ω (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18		ax-bdel 9941	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝑥
19	18	ax-bdn 9937	. . 3 ⊢ BOUNDED ¬ 𝑥 ∈ 𝑥
20		nfv 1421	. . 3 ⊢ Ⅎ𝑥 ¬ ∅ ∈ ∅
21		nfv 1421	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝑦
22		nfv 1421	. . 3 ⊢ Ⅎ𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23		eleq1 2100	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
24		eleq2 2101	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
25	23, 24	bitrd 177	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑥 ↔ ∅ ∈ ∅))
26	25	notbid 592	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ 𝑥 ↔ ¬ ∅ ∈ ∅))
27	26	biimprd 147	. . 3 ⊢ (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥 ∈ 𝑥))
28		elequ1 1600	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
29		elequ2 1601	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
30	28, 29	bitrd 177	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
31	30	notbid 592	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
32	31	biimpd 132	. . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑦))
33		eleq1 2100	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝑥 ↔ suc 𝑦 ∈ 𝑥))
34		eleq2 2101	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ∈ 𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
35	33, 34	bitrd 177	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ∈ 𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
36	35	notbid 592	. . . 4 ⊢ (𝑥 = suc 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
37	36	biimprd 147	. . 3 ⊢ (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥 ∈ 𝑥))
38		nfcv 2178	. . 3 ⊢ Ⅎ𝑥𝐴
39		nfv 1421	. . 3 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐴
40		eleq1 2100	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
41		eleq2 2101	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
42	40, 41	bitrd 177	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
43	42	notbid 592	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
44	43	biimpd 132	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴))
45	19, 20, 21, 22, 27, 32, 37, 38, 39, 44	bj-bdfindisg 10073	. 2 ⊢ ((¬ ∅ ∈ ∅ ∧ ∀𝑦 ∈ ω (¬ 𝑦 ∈ 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)) → (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴))
46	1, 17, 45	mp2an 402	1 ⊢ (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ∪ cun 2915   ⊆ wss 2917  ∅c0 3224  {csn 3375  suc csuc 4102  ωcom 4313

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdn 9937  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051

This theorem is referenced by:  bj-nnen2lp  10079