Theorem oeworde 8524

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 22	. . . 4 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
2		oveq2 7370	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
3	1, 2	sseq12d 3956	. . 3 ⊢ (𝑥 = ∅ → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ ∅ ⊆ (𝐴 ↑o ∅)))
4		id 22	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
5		oveq2 7370	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
6	4, 5	sseq12d 3956	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ 𝑦 ⊆ (𝐴 ↑o 𝑦)))
7		id 22	. . . 4 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
8		oveq2 7370	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝑦))
9	7, 8	sseq12d 3956	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
11		oveq2 7370	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝐵))
12	10, 11	sseq12d 3956	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ 𝐵 ⊆ (𝐴 ↑o 𝐵)))
13		0ss 4341	. . . 4 ⊢ ∅ ⊆ (𝐴 ↑o ∅)
14	13	a1i 11	. . 3 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ⊆ (𝐴 ↑o ∅))
15		eloni 6329	. . . . . 6 ⊢ (𝑦 ∈ On → Ord 𝑦)
16		eldifi 4072	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
17		oecl 8467	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
18	16, 17	sylan 581	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ On)
19		eloni 6329	. . . . . . 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 7769	. . . . . 6 ⊢ ((Ord 𝑦 ∧ Ord (𝐴 ↑o 𝑦)) → (𝑦 ⊆ (𝐴 ↑o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦)))
22	15, 20, 21	syl2an2 687	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑o 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑o 𝑦)))
23		onsuc 7759	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
24		oecl 8467	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) ∈ On)
25	16, 23, 24	syl2an 597	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o suc 𝑦) ∈ On)
26		eloni 6329	. . . . . . . 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 3434	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
30	29	sucid 6403	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
31		oeordi 8518	. . . . . . . . 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 598	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝐴 ↑o 𝑦) ∈ (𝐴 ↑o suc 𝑦))
34		ordsucss 7764	. . . . . . 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 3929	. . . . . 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 240	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑o 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦)))
39	38	expcom 413	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2o) → (𝑦 ⊆ (𝐴 ↑o 𝑦) → suc 𝑦 ⊆ (𝐴 ↑o suc 𝑦))))
40		dif20el 8435	. . . . 5 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
41	16, 40	jca 511	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2o) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
42		ss2iun 4953	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
43		limuni 6381	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
44		uniiun 5002	. . . . . . . . 9 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
45	43, 44	eqtrdi 2788	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
46	45	adantr 480	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
47		vex 3434	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
48		oelim 8464	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
49	47, 48	mpanlr1 707	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
50	49	anasss 466	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
51	50	an12s 650	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦))
52	46, 51	sseq12d 3956	. . . . . 6 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴 ↑o 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑o 𝑦)))
53	42, 52	imbitrrid 246	. . . . 5 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑o 𝑦) → 𝑥 ⊆ (𝐴 ↑o 𝑥)))
54	53	ex 412	. . . 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 7811	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2o) → 𝐵 ⊆ (𝐴 ↑o 𝐵)))
57	56	impcom 407	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  ∪ cuni 4851  ∪ ciun 4934  Ord word 6318  Oncon0 6319  Lim wlim 6320  suc csuc 6321  (class class class)co 7362  2_oc2o 8394   ↑_o coe 8399

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-omul 8405  df-oexp 8406

This theorem is referenced by:  oeeulem  8532  cnfcom3clem  9621  oege2  43757  nnoeomeqom  43762