Theorem bj-omtrans 14848

Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4607.

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 14834	. . 3 ⊢ ω ∈ V
2		sseq2 3181	. . . . . 6 ⊢ (𝑎 = ω → (𝑦 ⊆ 𝑎 ↔ 𝑦 ⊆ ω))
3		sseq2 3181	. . . . . 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 2477	. . . 4 ⊢ (𝑎 = ω → (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6		sseq2 3181	. . . . 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 3463	. . . 4 ⊢ ∅ ⊆ 𝑎
10		bdcv 14740	. . . . . 6 ⊢ BOUNDED 𝑎
11	10	bdss 14756	. . . . 5 ⊢ BOUNDED 𝑥 ⊆ 𝑎
12		nfv 1528	. . . . 5 ⊢ Ⅎ𝑥∅ ⊆ 𝑎
13		nfv 1528	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑎
14		nfv 1528	. . . . 5 ⊢ Ⅎ𝑥 suc 𝑦 ⊆ 𝑎
15		sseq1 3180	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑎 ↔ ∅ ⊆ 𝑎))
16	15	biimprd 158	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
17		sseq1 3180	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 ↔ 𝑦 ⊆ 𝑎))
18	17	biimpd 144	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 → 𝑦 ⊆ 𝑎))
19		sseq1 3180	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ 𝑎 ↔ suc 𝑦 ⊆ 𝑎))
20	19	biimprd 158	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
21		nfcv 2319	. . . . 5 ⊢ Ⅎ𝑥𝐴
22		nfv 1528	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑎
23		sseq1 3180	. . . . . 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 14840	. . . 4 ⊢ ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎)) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
26	9, 25	mpan 424	. . 3 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
27	1, 8, 26	vtocl 2793	. 2 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28		df-suc 4373	. . . 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 3739	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
32	29, 31	unssd 3313	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
33	28, 32	eqsstrid 3203	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
34	33	ex 115	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
35	27, 34	mprg 2534	1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ∀wral 2455   ∪ cun 3129   ⊆ wss 3131  ∅c0 3424  {csn 3594  suc csuc 4367  ωcom 4591

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4131  ax-pr 4211  ax-un 4435  ax-bd0 14705  ax-bdor 14708  ax-bdal 14710  ax-bdex 14711  ax-bdeq 14712  ax-bdel 14713  ax-bdsb 14714  ax-bdsep 14776  ax-infvn 14833

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592  df-bdc 14733  df-bj-ind 14819

This theorem is referenced by:  bj-omtrans2  14849  bj-nnord  14850  bj-nn0suc  14856