Theorem oacl2g 43343

Description: Closure law for ordinal addition. Here we show that ordinal addition is closed within the empty set or any ordinal power of omega. (Contributed by RP, 5-Jan-2025.)

Assertion

Ref	Expression
oacl2g	⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On))) → (𝐴 +o 𝐵) ∈ 𝐶)

Proof of Theorem oacl2g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2830	. . . . . 6 ⊢ (𝐶 = ∅ → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ ∅))
2		noel 4338	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
3	2	pm2.21i 119	. . . . . 6 ⊢ (𝐴 ∈ ∅ → (𝐴 +o 𝐵) ∈ 𝐶)
4	1, 3	biimtrdi 253	. . . . 5 ⊢ (𝐶 = ∅ → (𝐴 ∈ 𝐶 → (𝐴 +o 𝐵) ∈ 𝐶))
5	4	com12 32	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝐶 = ∅ → (𝐴 +o 𝐵) ∈ 𝐶))
6	5	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 = ∅ → (𝐴 +o 𝐵) ∈ 𝐶))
7		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)
8		simpl 482	. . . . . . . . . 10 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → 𝐶 = (ω ↑o 𝐷))
9		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → 𝐷 ∈ On)
10		omelon 9686	. . . . . . . . . . . 12 ⊢ ω ∈ On
11	9, 10	jctil 519	. . . . . . . . . . 11 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → (ω ∈ On ∧ 𝐷 ∈ On))
12		oecl 8575	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ↑o 𝐷) ∈ On)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → (ω ↑o 𝐷) ∈ On)
14	8, 13	eqeltrd 2841	. . . . . . . . 9 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → 𝐶 ∈ On)
15	14	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → 𝐶 ∈ On)
16		onelon 6409	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ On)
17	16	expcom 413	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐶 → (𝐶 ∈ On → 𝐴 ∈ On))
18	17	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐶 ∈ On → 𝐴 ∈ On))
19	18	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → (𝐶 ∈ On → 𝐴 ∈ On))
20	15, 19	jcai 516	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐴 ∈ On))
21		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → 𝐵 ∈ 𝐶)
22	21	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → 𝐵 ∈ 𝐶)
23		oaordi 8584	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶)))
24	20, 22, 23	sylc 65	. . . . . 6 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶))
25		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 +o 𝐶) = (𝐴 +o 𝐶))
26	25	eliuni 4997	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶)) → (𝐴 +o 𝐵) ∈ ∪ 𝑥 ∈ 𝐶 (𝑥 +o 𝐶))
27	7, 24, 26	syl2an2r 685	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → (𝐴 +o 𝐵) ∈ ∪ 𝑥 ∈ 𝐶 (𝑥 +o 𝐶))
28		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
29	8	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 ∈ 𝐶) → 𝐶 = (ω ↑o 𝐷))
30	28, 29	eleqtrd 2843	. . . . . . . . . 10 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ (ω ↑o 𝐷))
31	14	adantr 480	. . . . . . . . . 10 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 ∈ 𝐶) → 𝐶 ∈ On)
32	8	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → (ω ↑o 𝐷) = 𝐶)
33		ssid 4006	. . . . . . . . . . . 12 ⊢ 𝐶 ⊆ 𝐶
34	32, 33	eqsstrdi 4028	. . . . . . . . . . 11 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → (ω ↑o 𝐷) ⊆ 𝐶)
35	34	adantr 480	. . . . . . . . . 10 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 ∈ 𝐶) → (ω ↑o 𝐷) ⊆ 𝐶)
36		oaabs2 8687	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (ω ↑o 𝐷) ∧ 𝐶 ∈ On) ∧ (ω ↑o 𝐷) ⊆ 𝐶) → (𝑥 +o 𝐶) = 𝐶)
37	30, 31, 35, 36	syl21anc 838	. . . . . . . . 9 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 ∈ 𝐶) → (𝑥 +o 𝐶) = 𝐶)
38	37, 33	eqsstrdi 4028	. . . . . . . 8 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 ∈ 𝐶) → (𝑥 +o 𝐶) ⊆ 𝐶)
39	38	iunssd 5050	. . . . . . 7 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → ∪ 𝑥 ∈ 𝐶 (𝑥 +o 𝐶) ⊆ 𝐶)
40		peano1 7910	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
41		oen0 8624	. . . . . . . . . . 11 ⊢ (((ω ∈ On ∧ 𝐷 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 𝐷))
42	11, 40, 41	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → ∅ ∈ (ω ↑o 𝐷))
43	42, 32	eleqtrd 2843	. . . . . . . . 9 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → ∅ ∈ 𝐶)
44		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 = ∅) → 𝑥 = ∅)
45	44	oveq1d 7446	. . . . . . . . . . 11 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 = ∅) → (𝑥 +o 𝐶) = (∅ +o 𝐶))
46		oa0r 8576	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → (∅ +o 𝐶) = 𝐶)
47	14, 46	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → (∅ +o 𝐶) = 𝐶)
48	47	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 = ∅) → (∅ +o 𝐶) = 𝐶)
49	45, 48	eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 = ∅) → (𝑥 +o 𝐶) = 𝐶)
50	49	sseq2d 4016	. . . . . . . . 9 ⊢ (((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) ∧ 𝑥 = ∅) → (𝐶 ⊆ (𝑥 +o 𝐶) ↔ 𝐶 ⊆ 𝐶))
51		ssidd 4007	. . . . . . . . 9 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → 𝐶 ⊆ 𝐶)
52	43, 50, 51	rspcedvd 3624	. . . . . . . 8 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → ∃𝑥 ∈ 𝐶 𝐶 ⊆ (𝑥 +o 𝐶))
53		ssiun 5046	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐶 𝐶 ⊆ (𝑥 +o 𝐶) → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐶 (𝑥 +o 𝐶))
54	52, 53	syl 17	. . . . . . 7 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐶 (𝑥 +o 𝐶))
55	39, 54	eqssd 4001	. . . . . 6 ⊢ ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → ∪ 𝑥 ∈ 𝐶 (𝑥 +o 𝐶) = 𝐶)
56	55	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → ∪ 𝑥 ∈ 𝐶 (𝑥 +o 𝐶) = 𝐶)
57	27, 56	eleqtrd 2843	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → (𝐴 +o 𝐵) ∈ 𝐶)
58	57	ex 412	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On) → (𝐴 +o 𝐵) ∈ 𝐶))
59	6, 58	jaod 860	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → ((𝐶 = ∅ ∨ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On)) → (𝐴 +o 𝐵) ∈ 𝐶))
60	59	imp 406	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On))) → (𝐴 +o 𝐵) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333  ∪ ciun 4991  Oncon0 6384  (class class class)co 7431  ωcom 7887   +_o coa 8503   ↑_o coe 8505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512

This theorem is referenced by:  onmcl  43344