Theorem bj-omtrans 10440

Description: The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4355.

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 10426	. . 3 ⊢ ω ∈ V
2		sseq2 2994	. . . . . 6 ⊢ (𝑎 = ω → (𝑦 ⊆ 𝑎 ↔ 𝑦 ⊆ ω))
3		sseq2 2994	. . . . . 6 ⊢ (𝑎 = ω → (suc 𝑦 ⊆ 𝑎 ↔ suc 𝑦 ⊆ ω))
4	2, 3	imbi12d 227	. . . . 5 ⊢ (𝑎 = ω → ((𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) ↔ (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
5	4	ralbidv 2343	. . . 4 ⊢ (𝑎 = ω → (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) ↔ ∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω)))
6		sseq2 2994	. . . . 5 ⊢ (𝑎 = ω → (𝐴 ⊆ 𝑎 ↔ 𝐴 ⊆ ω))
7	6	imbi2d 223	. . . 4 ⊢ (𝑎 = ω → ((𝐴 ∈ ω → 𝐴 ⊆ 𝑎) ↔ (𝐴 ∈ ω → 𝐴 ⊆ ω)))
8	5, 7	imbi12d 227	. . 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 3282	. . . 4 ⊢ ∅ ⊆ 𝑎
10		bdcv 10327	. . . . . 6 ⊢ BOUNDED 𝑎
11	10	bdss 10343	. . . . 5 ⊢ BOUNDED 𝑥 ⊆ 𝑎
12		nfv 1437	. . . . 5 ⊢ Ⅎ𝑥∅ ⊆ 𝑎
13		nfv 1437	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑎
14		nfv 1437	. . . . 5 ⊢ Ⅎ𝑥 suc 𝑦 ⊆ 𝑎
15		sseq1 2993	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑎 ↔ ∅ ⊆ 𝑎))
16	15	biimprd 151	. . . . 5 ⊢ (𝑥 = ∅ → (∅ ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
17		sseq1 2993	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 ↔ 𝑦 ⊆ 𝑎))
18	17	biimpd 136	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑎 → 𝑦 ⊆ 𝑎))
19		sseq1 2993	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ 𝑎 ↔ suc 𝑦 ⊆ 𝑎))
20	19	biimprd 151	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝑦 ⊆ 𝑎 → 𝑥 ⊆ 𝑎))
21		nfcv 2194	. . . . 5 ⊢ Ⅎ𝑥𝐴
22		nfv 1437	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝑎
23		sseq1 2993	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑎 ↔ 𝐴 ⊆ 𝑎))
24	23	biimpd 136	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑎 → 𝐴 ⊆ 𝑎))
25	11, 12, 13, 14, 16, 18, 20, 21, 22, 24	bj-bdfindisg 10432	. . . 4 ⊢ ((∅ ⊆ 𝑎 ∧ ∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎)) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
26	9, 25	mpan 408	. . 3 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ 𝑎 → suc 𝑦 ⊆ 𝑎) → (𝐴 ∈ ω → 𝐴 ⊆ 𝑎))
27	1, 8, 26	vtocl 2625	. 2 ⊢ (∀𝑦 ∈ ω (𝑦 ⊆ ω → suc 𝑦 ⊆ ω) → (𝐴 ∈ ω → 𝐴 ⊆ ω))
28		df-suc 4135	. . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
29		simpr 107	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ⊆ ω)
30		simpl 106	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → 𝑦 ∈ ω)
31	30	snssd 3536	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → {𝑦} ⊆ ω)
32	29, 31	unssd 3146	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → (𝑦 ∪ {𝑦}) ⊆ ω)
33	28, 32	syl5eqss 3016	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ⊆ ω) → suc 𝑦 ⊆ ω)
34	33	ex 112	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ⊆ ω → suc 𝑦 ⊆ ω))
35	27, 34	mprg 2395	1 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409  ∀wral 2323   ∪ cun 2942   ⊆ wss 2944  ∅c0 3251  {csn 3402  suc csuc 4129  ωcom 4340

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910  ax-pr 3971  ax-un 4197  ax-bd0 10292  ax-bdor 10295  ax-bdal 10297  ax-bdex 10298  ax-bdeq 10299  ax-bdel 10300  ax-bdsb 10301  ax-bdsep 10363  ax-infvn 10425

This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-pr 3409  df-uni 3608  df-int 3643  df-suc 4135  df-iom 4341  df-bdc 10320  df-bj-ind 10410

This theorem is referenced by:  bj-omtrans2  10441  bj-nnord  10442  bj-nn0suc  10448