Theorem bj-omtrans 16896

Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elomssom 4747.

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 16882	. . 3 ⊢ ω ∈ V
2		sseq2 3272	. . . . . 6 ⊢ (𝑎 = ω → (𝑦 ⊆ 𝑎 ↔ 𝑦 ⊆ ω))
3		sseq2 3272	. . . . . 6 ⊢ (𝑎 = ω → (suc 𝑦 ⊆ 𝑎 ↔ suc 𝑦 ⊆ ω))
4	2, 3	imbi12d 234	. . . . 5 ⊢ (𝑎 = ω → ((𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) ↔ (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
5	4	ralbidv 2550	. . . 4 ⊢ (𝑎 = ω → (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6		sseq2 3272	. . . . 5 ⊢ (𝑎 = ω → (𝐴 ⊆ 𝑎 ↔ 𝐴 ⊆ ω))
7	6	imbi2d 230	. . . 4 ⊢ (𝑎 = ω → ((𝐴 ∈ ω → 𝐴 ⊆ 𝑎) ↔ (𝐴 ∈ ω → 𝐴 ⊆ ω)))
8	5, 7	imbi12d 234	. . 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 3561	. . . 4 ⊢ ∅ ⊆ 𝑎
10		bdcv 16788	. . . . . 6 ⊢ BOUNDED 𝑎
11	10	bdss 16804	. . . . 5 ⊢ BOUNDED 𝑥 ⊆ 𝑎
12		nfv 1581	. . . . 5 ⊢ Ⅎ𝑥∅ ⊆ 𝑎
13		nfv 1581	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑎
14		nfv 1581	. . . . 5 ⊢ Ⅎ𝑥 suc 𝑦 ⊆ 𝑎
15		sseq1 3271	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑎 ↔ ∅ ⊆ 𝑎))
16	15	biimprd 158	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
17		sseq1 3271	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 ↔ 𝑦 ⊆ 𝑎))
18	17	biimpd 144	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 → 𝑦 ⊆ 𝑎))
19		sseq1 3271	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ 𝑎 ↔ suc 𝑦 ⊆ 𝑎))
20	19	biimprd 158	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
21		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑥𝐴
22		nfv 1581	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑎
23		sseq1 3271	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑎 ↔ 𝐴 ⊆ 𝑎))
24	23	biimpd 144	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑎 → 𝐴 ⊆ 𝑎))
25	11, 12, 13, 14, 16, 18, 20, 21, 22, 24	bj-bdfindisg 16888	. . . 4 ⊢ ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎)) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
26	9, 25	mpan 428	. . 3 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
27	1, 8, 26	vtocl 2877	. 2 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28		df-suc 4511	. . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
29		simpr 110	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ⊆ ω)
30		simpl 109	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ∈ ω)
31	30	snssd 3855	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
32	29, 31	unssd 3405	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
33	28, 32	eqsstrid 3294	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
34	33	ex 115	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
35	27, 34	mprg 2607	1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  {csn 3705  suc csuc 4505  ωcom 4732

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-nul 4254  ax-pr 4341  ax-un 4573  ax-bd0 16753  ax-bdor 16756  ax-bdal 16758  ax-bdex 16759  ax-bdeq 16760  ax-bdel 16761  ax-bdsb 16762  ax-bdsep 16824  ax-infvn 16881

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733  df-bdc 16781  df-bj-ind 16867

This theorem is referenced by:  bj-omtrans2  16897  bj-nnord  16898  bj-nn0suc  16904