Theorem bj-nntrans 13497

Description: A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-nntrans	⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Proof of Theorem bj-nntrans

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

Step	Hyp	Ref	Expression
1		ral0 3495	. . 3 ⊢ ∀𝑥 ∈ ∅ 𝑥 ⊆ ∅
2		df-suc 4331	. . . . . . 7 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
3	2	eleq2i 2224	. . . . . 6 ⊢ (𝑥 ∈ suc 𝑧 ↔ 𝑥 ∈ (𝑧 ∪ {𝑧}))
4		elun 3248	. . . . . . 7 ⊢ (𝑥 ∈ (𝑧 ∪ {𝑧}) ↔ (𝑥 ∈ 𝑧 ∨ 𝑥 ∈ {𝑧}))
5		sssucid 4375	. . . . . . . . . 10 ⊢ 𝑧 ⊆ suc 𝑧
6		sstr2 3135	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑧 → (𝑧 ⊆ suc 𝑧 → 𝑥 ⊆ suc 𝑧))
7	5, 6	mpi 15	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝑧 → 𝑥 ⊆ suc 𝑧)
8	7	imim2i 12	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → (𝑥 ∈ 𝑧 → 𝑥 ⊆ suc 𝑧))
9		elsni 3578	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
10	9, 5	eqsstrdi 3180	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧)
11	10	a1i 9	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧))
12	8, 11	jaod 707	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → ((𝑥 ∈ 𝑧 ∨ 𝑥 ∈ {𝑧}) → 𝑥 ⊆ suc 𝑧))
13	4, 12	syl5bi 151	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → (𝑥 ∈ (𝑧 ∪ {𝑧}) → 𝑥 ⊆ suc 𝑧))
14	3, 13	syl5bi 151	. . . . 5 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → (𝑥 ∈ suc 𝑧 → 𝑥 ⊆ suc 𝑧))
15	14	ralimi2 2517	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
16	15	rgenw 2512	. . 3 ⊢ ∀𝑧 ∈ ω (∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
17		bdcv 13394	. . . . . 6 ⊢ BOUNDED 𝑦
18	17	bdss 13410	. . . . 5 ⊢ BOUNDED 𝑥 ⊆ 𝑦
19	18	ax-bdal 13364	. . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦
20		nfv 1508	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ ∅ 𝑥 ⊆ ∅
21		nfv 1508	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧
22		nfv 1508	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧
23		sseq2 3152	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ ∅))
24	23	raleqbi1dv 2660	. . . . 5 ⊢ (𝑦 = ∅ → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ ∅ 𝑥 ⊆ ∅))
25	24	biimprd 157	. . . 4 ⊢ (𝑦 = ∅ → (∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ → ∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦))
26		sseq2 3152	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑧))
27	26	raleqbi1dv 2660	. . . . 5 ⊢ (𝑦 = 𝑧 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧))
28	27	biimpd 143	. . . 4 ⊢ (𝑦 = 𝑧 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧))
29		sseq2 3152	. . . . . 6 ⊢ (𝑦 = suc 𝑧 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ suc 𝑧))
30	29	raleqbi1dv 2660	. . . . 5 ⊢ (𝑦 = suc 𝑧 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
31	30	biimprd 157	. . . 4 ⊢ (𝑦 = suc 𝑧 → (∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧 → ∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦))
32		nfcv 2299	. . . 4 ⊢ Ⅎ𝑦𝐴
33		nfv 1508	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴
34		sseq2 3152	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴))
35	34	raleqbi1dv 2660	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴))
36	35	biimpd 143	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴))
37	19, 20, 21, 22, 25, 28, 31, 32, 33, 36	bj-bdfindisg 13494	. . 3 ⊢ ((∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ ∧ ∀𝑧 ∈ ω (∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)) → (𝐴 ∈ ω → ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴))
38	1, 16, 37	mp2an 423	. 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
39		nfv 1508	. . 3 ⊢ Ⅎ𝑥 𝐵 ⊆ 𝐴
40		sseq1 3151	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
41	39, 40	rspc 2810	. 2 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → 𝐵 ⊆ 𝐴))
42	38, 41	syl5com 29	1 ⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∨ wo 698   = wceq 1335   ∈ wcel 2128  ∀wral 2435   ∪ cun 3100   ⊆ wss 3102  ∅c0 3394  {csn 3560  suc csuc 4325  ωcom 4548

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4090  ax-pr 4169  ax-un 4393  ax-bd0 13359  ax-bdor 13362  ax-bdal 13364  ax-bdex 13365  ax-bdeq 13366  ax-bdel 13367  ax-bdsb 13368  ax-bdsep 13430  ax-infvn 13487

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-suc 4331  df-iom 4549  df-bdc 13387  df-bj-ind 13473

This theorem is referenced by:  bj-nntrans2  13498  bj-nnelirr  13499  bj-nnen2lp  13500