Theorem oeworde 8234

Description: Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Assertion

Ref	Expression
oeworde	⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

Proof of Theorem oeworde

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
2		oveq2 7163	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
3	1, 2	sseq12d 3927	. . 3 ⊢ (𝑥 = ∅ → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ ∅ ⊆ (𝐴 ↑o ∅)))
4		id 22	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
5		oveq2 7163	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
6	4, 5	sseq12d 3927	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ 𝑦 ⊆ (𝐴 ↑o 𝑦)))
7		id 22	. . . 4 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
8		oveq2 7163	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
9	7, 8	sseq12d 3927	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
11		oveq2 7163	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
12	10, 11	sseq12d 3927	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ 𝐵 ⊆ (𝐴 ↑o 𝐵)))
13		0ss 4295	. . . 4 ⊢ ∅ ⊆ (𝐴 ↑o ∅)
14	13	a1i 11	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴 ↑o ∅))
15		eloni 6183	. . . . . 6 ⊢ (𝑦 ∈ On → Ord 𝑦)
16		eldifi 4034	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17		oecl 8177	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
18	16, 17	sylan 583	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
19		eloni 6183	. . . . . . 7 ⊢ ((𝐴 ↑o 𝑦) ∈ On → Ord (𝐴 ↑o 𝑦))
20	18, 19	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴 ↑o 𝑦))
21		ordsucsssuc 7542	. . . . . 6 ⊢ ((Ord 𝑦 ∧ Ord (𝐴 ↑o 𝑦)) → (𝑦 ⊆ (𝐴 ↑o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦)))
22	15, 20, 21	syl2an2 685	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦)))
23		suceloni 7532	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
24		oecl 8177	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) ∈ On)
25	16, 23, 24	syl2an 598	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) ∈ On)
26		eloni 6183	. . . . . . . 8 ⊢ ((𝐴 ↑o suc 𝑦) ∈ On → Ord (𝐴 ↑o suc 𝑦))
27	25, 26	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴 ↑o suc 𝑦))
28		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ (On ∖ 2o))
29		vex 3413	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
30	29	sucid 6252	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
31		oeordi 8228	. . . . . . . . 9 ⊢ ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ suc 𝑦 → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦)))
32	30, 31	mpi 20	. . . . . . . 8 ⊢ ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦))
33	23, 28, 32	syl2anr 599	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦))
34		ordsucss 7537	. . . . . . 7 ⊢ (Ord (𝐴 ↑o suc 𝑦) → ((𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦) → suc (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦)))
35	27, 33, 34	sylc 65	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → suc (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦))
36		sstr2 3901	. . . . . 6 ⊢ (suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦) → (suc (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
37	35, 36	syl5com 31	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
38	22, 37	sylbid 243	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑o 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
39	38	expcom 417	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴 ↑o 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦))))
40		dif20el 8145	. . . . 5 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
41	16, 40	jca 515	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42		ss2iun 4904	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43		limuni 6233	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
44		uniiun 4950	. . . . . . . . 9 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
45	43, 44	eqtrdi 2809	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
46	45	adantr 484	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
47		vex 3413	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
48		oelim 8174	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
49	47, 48	mpanlr1 705	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
50	49	anasss 470	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
51	50	an12s 648	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
52	46, 51	sseq12d 3927	. . . . . 6 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)))
53	42, 52	syl5ibr 249	. . . . 5 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → 𝑥 ⊆ (𝐴 ↑o 𝑥)))
54	53	ex 416	. . . 4 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → 𝑥 ⊆ (𝐴 ↑o 𝑥))))
55	41, 54	syl5 34	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ (On ∖ 2o) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → 𝑥 ⊆ (𝐴 ↑o 𝑥))))
56	3, 6, 9, 12, 14, 39, 55	tfinds3 7583	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2o) → 𝐵 ⊆ (𝐴 ↑o 𝐵)))
57	56	impcom 411	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∖ cdif 3857   ⊆ wss 3860  ∅c0 4227  ∪ cuni 4801  ∪ ciun 4886  Ord word 6172  Oncon0 6173  Lim wlim 6174  suc csuc 6175  (class class class)co 7155  2_oc2o 8111   ↑_o coe 8116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-2o 8118  df-oadd 8121  df-omul 8122  df-oexp 8123

This theorem is referenced by:  oeeulem  8242  cnfcom3clem  9206