Theorem oewordi 8422

Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)

Assertion

Ref	Expression
oewordi	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Proof of Theorem oewordi

Step	Hyp	Ref	Expression
1		eloni 6276	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordgt0ge1 8323	. . . . . 6 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
4		1on 8309	. . . . . 6 ⊢ 1o ∈ On
5		onsseleq 6307	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐶 ∈ On) → (1o ⊆ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
6	4, 5	mpan 687	. . . . 5 ⊢ (𝐶 ∈ On → (1o ⊆ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
7	3, 6	bitrd 278	. . . 4 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
8	7	3ad2ant3 1134	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
9		ondif2 8332	. . . . . . 7 ⊢ (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o ∈ 𝐶))
10		oeword 8421	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
11	10	biimpd 228	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
12	11	3expia 1120	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2o) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
13	9, 12	syl5bir 242	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 1o ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
14	13	expd 416	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1o ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))))
15	14	3impia 1116	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
16		oe1m 8376	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)
17	16	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = 1o)
18		oe1m 8376	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o)
19	18	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐵) = 1o)
20	17, 19	eqtr4d 2781	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = (1o ↑o 𝐵))
21		eqimss 3977	. . . . . . . 8 ⊢ ((1o ↑o 𝐴) = (1o ↑o 𝐵) → (1o ↑o 𝐴) ⊆ (1o ↑o 𝐵))
22	20, 21	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) ⊆ (1o ↑o 𝐵))
23		oveq1 7282	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐴) = (𝐶 ↑o 𝐴))
24		oveq1 7282	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐵) = (𝐶 ↑o 𝐵))
25	23, 24	sseq12d 3954	. . . . . . 7 ⊢ (1o = 𝐶 → ((1o ↑o 𝐴) ⊆ (1o ↑o 𝐵) ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
26	22, 25	syl5ibcom 244	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o = 𝐶 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
27	26	3adant3 1131	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o = 𝐶 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))
28	27	a1dd 50	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o = 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
29	15, 28	jaod 856	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((1o ∈ 𝐶 ∨ 1o = 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
30	8, 29	sylbid 239	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
31	30	imp 407	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  Ord word 6265  Oncon0 6266  (class class class)co 7275  1_oc1o 8290  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:  oelim2  8426  oeoalem  8427  oeoelem  8429  oaabs2  8479  cantnflt  9430  cnfcom  9458