Theorem bj-omtrans 13143

Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4514.

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 13129	. . 3 ⊢ ω ∈ V
2		sseq2 3116	. . . . . 6 ⊢ (𝑎 = ω → (𝑦 ⊆ 𝑎 ↔ 𝑦 ⊆ ω))
3		sseq2 3116	. . . . . 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 2435	. . . 4 ⊢ (𝑎 = ω → (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6		sseq2 3116	. . . . 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 3396	. . . 4 ⊢ ∅ ⊆ 𝑎
10		bdcv 13035	. . . . . 6 ⊢ BOUNDED 𝑎
11	10	bdss 13051	. . . . 5 ⊢ BOUNDED 𝑥 ⊆ 𝑎
12		nfv 1508	. . . . 5 ⊢ Ⅎ𝑥∅ ⊆ 𝑎
13		nfv 1508	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑎
14		nfv 1508	. . . . 5 ⊢ Ⅎ𝑥 suc 𝑦 ⊆ 𝑎
15		sseq1 3115	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑎 ↔ ∅ ⊆ 𝑎))
16	15	biimprd 157	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
17		sseq1 3115	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 ↔ 𝑦 ⊆ 𝑎))
18	17	biimpd 143	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 → 𝑦 ⊆ 𝑎))
19		sseq1 3115	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ 𝑎 ↔ suc 𝑦 ⊆ 𝑎))
20	19	biimprd 157	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
21		nfcv 2279	. . . . 5 ⊢ Ⅎ𝑥𝐴
22		nfv 1508	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑎
23		sseq1 3115	. . . . . 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 13135	. . . 4 ⊢ ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎)) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
26	9, 25	mpan 420	. . 3 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
27	1, 8, 26	vtocl 2735	. 2 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28		df-suc 4288	. . . 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 3660	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
32	29, 31	unssd 3247	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
33	28, 32	eqsstrid 3138	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
34	33	ex 114	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
35	27, 34	mprg 2487	1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2414   ∪ cun 3064   ⊆ wss 3066  ∅c0 3358  {csn 3522  suc csuc 4282  ωcom 4499

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049  ax-pr 4126  ax-un 4350  ax-bd0 13000  ax-bdor 13003  ax-bdal 13005  ax-bdex 13006  ax-bdeq 13007  ax-bdel 13008  ax-bdsb 13009  ax-bdsep 13071  ax-infvn 13128

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-suc 4288  df-iom 4500  df-bdc 13028  df-bj-ind 13114

This theorem is referenced by:  bj-omtrans2  13144  bj-nnord  13145  bj-nn0suc  13151