Theorem bj-nntrans 16086

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 3570	. . 3 ⊢ ∀𝑥 ∈ ∅ 𝑥 ⊆ ∅
2		df-suc 4436	. . . . . . 7 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
3	2	eleq2i 2274	. . . . . 6 ⊢ (𝑥 ∈ suc 𝑧 ↔ 𝑥 ∈ (𝑧 ∪ {𝑧}))
4		elun 3322	. . . . . . 7 ⊢ (𝑥 ∈ (𝑧 ∪ {𝑧}) ↔ (𝑥 ∈ 𝑧 ∨ 𝑥 ∈ {𝑧}))
5		sssucid 4480	. . . . . . . . . 10 ⊢ 𝑧 ⊆ suc 𝑧
6		sstr2 3208	. . . . . . . . . 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 3661	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
10	9, 5	eqsstrdi 3253	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧)
11	10	a1i 9	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → (𝑥 ∈ {𝑧} → 𝑥 ⊆ suc 𝑧))
12	8, 11	jaod 719	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → ((𝑥 ∈ 𝑧 ∨ 𝑥 ∈ {𝑧}) → 𝑥 ⊆ suc 𝑧))
13	4, 12	biimtrid 152	. . . . . 6 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → (𝑥 ∈ (𝑧 ∪ {𝑧}) → 𝑥 ⊆ suc 𝑧))
14	3, 13	biimtrid 152	. . . . 5 ⊢ ((𝑥 ∈ 𝑧 → 𝑥 ⊆ 𝑧) → (𝑥 ∈ suc 𝑧 → 𝑥 ⊆ suc 𝑧))
15	14	ralimi2 2568	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
16	15	rgenw 2563	. . 3 ⊢ ∀𝑧 ∈ ω (∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧 → ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧)
17		bdcv 15983	. . . . . 6 ⊢ BOUNDED 𝑦
18	17	bdss 15999	. . . . 5 ⊢ BOUNDED 𝑥 ⊆ 𝑦
19	18	ax-bdal 15953	. . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦
20		nfv 1552	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ ∅ 𝑥 ⊆ ∅
21		nfv 1552	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧
22		nfv 1552	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧
23		sseq2 3225	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ ∅))
24	23	raleqbi1dv 2717	. . . . 5 ⊢ (𝑦 = ∅ → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ ∅ 𝑥 ⊆ ∅))
25	24	biimprd 158	. . . 4 ⊢ (𝑦 = ∅ → (∀𝑥 ∈ ∅ 𝑥 ⊆ ∅ → ∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦))
26		sseq2 3225	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑧))
27	26	raleqbi1dv 2717	. . . . 5 ⊢ (𝑦 = 𝑧 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧))
28	27	biimpd 144	. . . 4 ⊢ (𝑦 = 𝑧 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝑧 𝑥 ⊆ 𝑧))
29		sseq2 3225	. . . . . 6 ⊢ (𝑦 = suc 𝑧 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ suc 𝑧))
30	29	raleqbi1dv 2717	. . . . 5 ⊢ (𝑦 = suc 𝑧 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧))
31	30	biimprd 158	. . . 4 ⊢ (𝑦 = suc 𝑧 → (∀𝑥 ∈ suc 𝑧𝑥 ⊆ suc 𝑧 → ∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦))
32		nfcv 2350	. . . 4 ⊢ Ⅎ𝑦𝐴
33		nfv 1552	. . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴
34		sseq2 3225	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴))
35	34	raleqbi1dv 2717	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴))
36	35	biimpd 144	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 𝑥 ⊆ 𝑦 → ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴))
37	19, 20, 21, 22, 25, 28, 31, 32, 33, 36	bj-bdfindisg 16083	. . 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 426	. 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
39		nfv 1552	. . 3 ⊢ Ⅎ𝑥 𝐵 ⊆ 𝐴
40		sseq1 3224	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
41	39, 40	rspc 2878	. 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 710   = wceq 1373   ∈ wcel 2178  ∀wral 2486   ∪ cun 3172   ⊆ wss 3174  ∅c0 3468  {csn 3643  suc csuc 4430  ωcom 4656

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-nul 4186  ax-pr 4269  ax-un 4498  ax-bd0 15948  ax-bdor 15951  ax-bdal 15953  ax-bdex 15954  ax-bdeq 15955  ax-bdel 15956  ax-bdsb 15957  ax-bdsep 16019  ax-infvn 16076

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649  df-pr 3650  df-uni 3865  df-int 3900  df-suc 4436  df-iom 4657  df-bdc 15976  df-bj-ind 16062

This theorem is referenced by:  bj-nntrans2  16087  bj-nnelirr  16088  bj-nnen2lp  16089