Theorem oeworde 8581

Description: Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. Lemma 3.20 of [Schloeder] p. 10. (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 23	. . . 4 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
2		oveq2 7417	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
3	1, 2	sseq12d 3964	. . 3 ⊢ (𝑥 = ∅ → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ ∅ ⊆ (𝐴 ↑o ∅)))
4		id 23	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
5		oveq2 7417	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
6	4, 5	sseq12d 3964	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ 𝑦 ⊆ (𝐴 ↑o 𝑦)))
7		id 23	. . . 4 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
8		oveq2 7417	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
9	7, 8	sseq12d 3964	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
10		id 23	. . . 4 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
11		oveq2 7417	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
12	10, 11	sseq12d 3964	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ 𝐵 ⊆ (𝐴 ↑o 𝐵)))
13		0ss 4350	. . . 4 ⊢ ∅ ⊆ (𝐴 ↑o ∅)
14	13	a1i 11	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴 ↑o ∅))
15		eloni 6362	. . . . . 6 ⊢ (𝑦 ∈ On → Ord 𝑦)
16		eldifi 4078	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17		oecl 8524	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
18	16, 17	sylan 592	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
19		eloni 6362	. . . . . . 7 ⊢ ((𝐴 ↑o 𝑦) ∈ On → Ord (𝐴 ↑o 𝑦))
20	18, 19	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴 ↑o 𝑦))
21		ordsucsssuc 7818	. . . . . 6 ⊢ ((Ord 𝑦 ∧ Ord (𝐴 ↑o 𝑦)) → (𝑦 ⊆ (𝐴 ↑o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦)))
22	15, 20, 21	syl2an2 699	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦)))
23		onsuc 7808	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
24		oecl 8524	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) ∈ On)
25	16, 23, 24	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) ∈ On)
26		eloni 6362	. . . . . . . 8 ⊢ ((𝐴 ↑o suc 𝑦) ∈ On → Ord (𝐴 ↑o suc 𝑦))
27	25, 26	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → Ord (𝐴 ↑o suc 𝑦))
28		id 23	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ (On ∖ 2o))
29		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
30	29	sucid 6437	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
31		oeordi 8575	. . . . . . . . 9 ⊢ ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑦 ∈ suc 𝑦 → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦)))
32	30, 31	mpi 21	. . . . . . . 8 ⊢ ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦))
33	23, 28, 32	syl2anr 609	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦))
34		ordsucss 7813	. . . . . . 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 66	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → suc (𝐴 ↑o 𝑦) ⊆ (𝐴 ↑o suc 𝑦))
36		sstr2 3938	. . . . . 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 32	. . . . 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 419	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴 ↑o 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦))))
40		dif20el 8492	. . . . 5 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
41	16, 40	jca 521	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42		ss2iun 4970	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43		limuni 6415	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
44		uniiun 5017	. . . . . . . . 9 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
45	43, 44	eqtrdi 2811	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
46	45	adantr 486	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
47		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
48		oelim 8521	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
49	47, 48	mpanlr1 719	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
50	49	anasss 472	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
51	50	an12s 662	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
52	46, 51	sseq12d 3964	. . . . . 6 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)))
53	42, 52	imbitrrid 249	. . . . 5 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → 𝑥 ⊆ (𝐴 ↑o 𝑥)))
54	53	ex 418	. . . 4 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → 𝑥 ⊆ (𝐴 ↑o 𝑥))))
55	41, 54	syl5 35	. . 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 7860	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2o) → 𝐵 ⊆ (𝐴 ↑o 𝐵)))
57	56	impcom 413	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867  ∪ ciun 4951  Ord word 6351  Oncon0 6352  Lim wlim 6353  suc csuc 6354  (class class class)co 7409  2_oc2o 8449   ↑_o coe 8454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7862  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8455  df-2o 8456  df-oadd 8459  df-omul 8460  df-oexp 8461

This theorem is used by:  oeeulem  8589  cnfcom3clem  9684  oege2  44246  nnoeomeqom  44251