Theorem oeworde 7673

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 6658	. . . 4 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
3	1, 2	sseq12d 3634	. . 3 ⊢ (𝑥 = ∅ → (𝑥 ⊆ (𝐴 ↑𝑜 𝑥) ↔ ∅ ⊆ (𝐴 ↑𝑜 ∅)))
4		id 22	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
5		oveq2 6658	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
6	4, 5	sseq12d 3634	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐴 ↑𝑜 𝑥) ↔ 𝑦 ⊆ (𝐴 ↑𝑜 𝑦)))
7		id 22	. . . 4 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
8		oveq2 6658	. . . 4 ⊢ (𝑥 = suc 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝑦))
9	7, 8	sseq12d 3634	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ (𝐴 ↑𝑜 𝑥) ↔ suc 𝑦 ⊆ (𝐴 ↑𝑜 suc 𝑦)))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
11		oveq2 6658	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝐵))
12	10, 11	sseq12d 3634	. . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐴 ↑𝑜 𝑥) ↔ 𝐵 ⊆ (𝐴 ↑𝑜 𝐵)))
13		0ss 3972	. . . 4 ⊢ ∅ ⊆ (𝐴 ↑𝑜 ∅)
14	13	a1i 11	. . 3 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ⊆ (𝐴 ↑𝑜 ∅))
15		eloni 5733	. . . . . . 7 ⊢ (𝑦 ∈ On → Ord 𝑦)
16	15	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord 𝑦)
17		eldifi 3732	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
18		oecl 7617	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
19	17, 18	sylan 488	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ On)
20		eloni 5733	. . . . . . 7 ⊢ ((𝐴 ↑𝑜 𝑦) ∈ On → Ord (𝐴 ↑𝑜 𝑦))
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord (𝐴 ↑𝑜 𝑦))
22		ordsucsssuc 7023	. . . . . 6 ⊢ ((Ord 𝑦 ∧ Ord (𝐴 ↑𝑜 𝑦)) → (𝑦 ⊆ (𝐴 ↑𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑𝑜 𝑦)))
23	16, 21, 22	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑𝑜 𝑦) ↔ suc 𝑦 ⊆ suc (𝐴 ↑𝑜 𝑦)))
24		suceloni 7013	. . . . . . . . 9 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
25		oecl 7617	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ suc 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) ∈ On)
26	17, 24, 25	syl2an 494	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 suc 𝑦) ∈ On)
27		eloni 5733	. . . . . . . 8 ⊢ ((𝐴 ↑𝑜 suc 𝑦) ∈ On → Ord (𝐴 ↑𝑜 suc 𝑦))
28	26, 27	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → Ord (𝐴 ↑𝑜 suc 𝑦))
29		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ (On ∖ 2𝑜))
30		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
31	30	sucid 5804	. . . . . . . . 9 ⊢ 𝑦 ∈ suc 𝑦
32		oeordi 7667	. . . . . . . . 9 ⊢ ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝑦 ∈ suc 𝑦 → (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 suc 𝑦)))
33	31, 32	mpi 20	. . . . . . . 8 ⊢ ((suc 𝑦 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 suc 𝑦))
34	24, 29, 33	syl2anr 495	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 suc 𝑦))
35		ordsucss 7018	. . . . . . 7 ⊢ (Ord (𝐴 ↑𝑜 suc 𝑦) → ((𝐴 ↑𝑜 𝑦) ∈ (𝐴 ↑𝑜 suc 𝑦) → suc (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦)))
36	28, 34, 35	sylc 65	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → suc (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦))
37		sstr2 3610	. . . . . 6 ⊢ (suc 𝑦 ⊆ suc (𝐴 ↑𝑜 𝑦) → (suc (𝐴 ↑𝑜 𝑦) ⊆ (𝐴 ↑𝑜 suc 𝑦) → suc 𝑦 ⊆ (𝐴 ↑𝑜 suc 𝑦)))
38	36, 37	syl5com 31	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (suc 𝑦 ⊆ suc (𝐴 ↑𝑜 𝑦) → suc 𝑦 ⊆ (𝐴 ↑𝑜 suc 𝑦)))
39	23, 38	sylbid 230	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝑦 ∈ On) → (𝑦 ⊆ (𝐴 ↑𝑜 𝑦) → suc 𝑦 ⊆ (𝐴 ↑𝑜 suc 𝑦)))
40	39	expcom 451	. . 3 ⊢ (𝑦 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → (𝑦 ⊆ (𝐴 ↑𝑜 𝑦) → suc 𝑦 ⊆ (𝐴 ↑𝑜 suc 𝑦))))
41		dif20el 7585	. . . . 5 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
42	17, 41	jca 554	. . . 4 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
43		ss2iun 4536	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑𝑜 𝑦) → ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
44		limuni 5785	. . . . . . . . 9 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑥)
45		uniiun 4573	. . . . . . . . 9 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
46	44, 45	syl6eq 2672	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
47	46	adantr 481	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦)
48		vex 3203	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
49		oelim 7614	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
50	48, 49	mpanlr1 722	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ Lim 𝑥) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
51	50	anasss 679	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (Lim 𝑥 ∧ ∅ ∈ 𝐴)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
52	51	an12s 843	. . . . . . 7 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝐴 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦))
53	47, 52	sseq12d 3634	. . . . . 6 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (𝑥 ⊆ (𝐴 ↑𝑜 𝑥) ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ⊆ ∪ 𝑦 ∈ 𝑥 (𝐴 ↑𝑜 𝑦)))
54	43, 53	syl5ibr 236	. . . . 5 ⊢ ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑𝑜 𝑦) → 𝑥 ⊆ (𝐴 ↑𝑜 𝑥)))
55	54	ex 450	. . . 4 ⊢ (Lim 𝑥 → ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑𝑜 𝑦) → 𝑥 ⊆ (𝐴 ↑𝑜 𝑥))))
56	42, 55	syl5 34	. . 3 ⊢ (Lim 𝑥 → (𝐴 ∈ (On ∖ 2𝑜) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐴 ↑𝑜 𝑦) → 𝑥 ⊆ (𝐴 ↑𝑜 𝑥))))
57	3, 6, 9, 12, 14, 40, 56	tfinds3 7064	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ (On ∖ 2𝑜) → 𝐵 ⊆ (𝐴 ↑𝑜 𝐵)))
58	57	impcom 446	1 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑𝑜 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436  ∪ ciun 4520  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650  2_𝑜c2o 7554   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oeeulem  7681  cnfcom3clem  8602