Theorem oeworde 8424

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 7283	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
3	1, 2	sseq12d 3954	. . 3 ⊢ (𝑥 = ∅ → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ ∅ ⊆ (𝐴 ↑o ∅)))
4		id 22	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
5		oveq2 7283	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
6	4, 5	sseq12d 3954	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ 𝑦 ⊆ (𝐴 ↑o 𝑦)))
7		id 22	. . . 4 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
8		oveq2 7283	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
9	7, 8	sseq12d 3954	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
11		oveq2 7283	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
12	10, 11	sseq12d 3954	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ 𝐵 ⊆ (𝐴 ↑o 𝐵)))
13		0ss 4330	. . . 4 ⊢ ∅ ⊆ (𝐴 ↑o ∅)
14	13	a1i 11	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴 ↑o ∅))
15		eloni 6276	. . . . . 6 ⊢ (𝑦 ∈ On → Ord 𝑦)
16		eldifi 4061	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17		oecl 8367	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
18	16, 17	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
19		eloni 6276	. . . . . . 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 7670	. . . . . 6 ⊢ ((Ord 𝑦 ∧ Ord (𝐴 ↑o 𝑦)) → (𝑦 ⊆ (𝐴 ↑o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦)))
22	15, 20, 21	syl2an2 683	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦)))
23		suceloni 7659	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
24		oecl 8367	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) ∈ On)
25	16, 23, 24	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) ∈ On)
26		eloni 6276	. . . . . . . 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 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
30	29	sucid 6345	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
31		oeordi 8418	. . . . . . . . 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 597	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦))
34		ordsucss 7665	. . . . . . 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 3928	. . . . . 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 239	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑o 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
39	38	expcom 414	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴 ↑o 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦))))
40		dif20el 8335	. . . . 5 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
41	16, 40	jca 512	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42		ss2iun 4942	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43		limuni 6326	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
44		uniiun 4988	. . . . . . . . 9 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
45	43, 44	eqtrdi 2794	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
46	45	adantr 481	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
47		vex 3436	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
48		oelim 8364	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
49	47, 48	mpanlr1 703	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
50	49	anasss 467	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
51	50	an12s 646	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
52	46, 51	sseq12d 3954	. . . . . 6 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)))
53	42, 52	syl5ibr 245	. . . . 5 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → 𝑥 ⊆ (𝐴 ↑o 𝑥)))
54	53	ex 413	. . . 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 7711	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2o) → 𝐵 ⊆ (𝐴 ↑o 𝐵)))
57	56	impcom 408	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839  ∪ ciun 4924  Ord word 6265  Oncon0 6266  Lim wlim 6267  suc csuc 6268  (class class class)co 7275  2_oc2o 8291   ↑_o coe 8296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-omul 8302  df-oexp 8303

This theorem is referenced by:  oeeulem  8432  cnfcom3clem  9463