Theorem bj-omtrans 13673

Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4577.

The idea is to use bounded induction with the formula 𝑥 ⊆ ω. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with 𝑥 ⊆ 𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-omtrans

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

Step	Hyp	Ref	Expression
1		bj-omex 13659	. . 3 ⊢ ω ∈ V
2		sseq2 3161	. . . . . 6 ⊢ (𝑎 = ω → (𝑦 ⊆ 𝑎 ↔ 𝑦 ⊆ ω))
3		sseq2 3161	. . . . . 6 ⊢ (𝑎 = ω → (suc 𝑦 ⊆ 𝑎 ↔ suc 𝑦 ⊆ ω))
4	2, 3	imbi12d 233	. . . . 5 ⊢ (𝑎 = ω → ((𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) ↔ (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
5	4	ralbidv 2464	. . . 4 ⊢ (𝑎 = ω → (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6		sseq2 3161	. . . . 5 ⊢ (𝑎 = ω → (𝐴 ⊆ 𝑎 ↔ 𝐴 ⊆ ω))
7	6	imbi2d 229	. . . 4 ⊢ (𝑎 = ω → ((𝐴 ∈ ω → 𝐴 ⊆ 𝑎) ↔ (𝐴 ∈ ω → 𝐴 ⊆ ω)))
8	5, 7	imbi12d 233	. . 3 ⊢ (𝑎 = ω → ((∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎)) ↔ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))))
9		0ss 3442	. . . 4 ⊢ ∅ ⊆ 𝑎
10		bdcv 13565	. . . . . 6 ⊢ BOUNDED 𝑎
11	10	bdss 13581	. . . . 5 ⊢ BOUNDED 𝑥 ⊆ 𝑎
12		nfv 1515	. . . . 5 ⊢ Ⅎ𝑥∅ ⊆ 𝑎
13		nfv 1515	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑎
14		nfv 1515	. . . . 5 ⊢ Ⅎ𝑥 suc 𝑦 ⊆ 𝑎
15		sseq1 3160	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑎 ↔ ∅ ⊆ 𝑎))
16	15	biimprd 157	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
17		sseq1 3160	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 ↔ 𝑦 ⊆ 𝑎))
18	17	biimpd 143	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 → 𝑦 ⊆ 𝑎))
19		sseq1 3160	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ 𝑎 ↔ suc 𝑦 ⊆ 𝑎))
20	19	biimprd 157	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
21		nfcv 2306	. . . . 5 ⊢ Ⅎ𝑥𝐴
22		nfv 1515	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑎
23		sseq1 3160	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑎 ↔ 𝐴 ⊆ 𝑎))
24	23	biimpd 143	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑎 → 𝐴 ⊆ 𝑎))
25	11, 12, 13, 14, 16, 18, 20, 21, 22, 24	bj-bdfindisg 13665	. . . 4 ⊢ ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎)) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
26	9, 25	mpan 421	. . 3 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
27	1, 8, 26	vtocl 2775	. 2 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28		df-suc 4343	. . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
29		simpr 109	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ⊆ ω)
30		simpl 108	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ∈ ω)
31	30	snssd 3712	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
32	29, 31	unssd 3293	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
33	28, 32	eqsstrid 3183	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
34	33	ex 114	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
35	27, 34	mprg 2521	1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1342   ∈ wcel 2135  ∀wral 2442   ∪ cun 3109   ⊆ wss 3111  ∅c0 3404  {csn 3570  suc csuc 4337  ωcom 4561

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-nul 4102  ax-pr 4181  ax-un 4405  ax-bd0 13530  ax-bdor 13533  ax-bdal 13535  ax-bdex 13536  ax-bdeq 13537  ax-bdel 13538  ax-bdsb 13539  ax-bdsep 13601  ax-infvn 13658

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-sn 3576  df-pr 3577  df-uni 3784  df-int 3819  df-suc 4343  df-iom 4562  df-bdc 13558  df-bj-ind 13644

This theorem is referenced by:  bj-omtrans2  13674  bj-nnord  13675  bj-nn0suc  13681