Theorem oewordi 8593

Description: Weak ordering property of ordinal exponentiation. Lemma 3.19 of [Schloeder] p. 10. (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 6373	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordgt0ge1 8495	. . . . . 6 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o ⊆ 𝐶))
4		1on 8480	. . . . . 6 ⊢ 1o ∈ On
5		onsseleq 6404	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐶 ∈ On) → (1o ⊆ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
6	4, 5	mpan 686	. . . . 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 1133	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1o ∈ 𝐶 ∨ 1o = 𝐶)))
9		ondif2 8504	. . . . . . 7 ⊢ (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o ∈ 𝐶))
10		oeword 8592	. . . . . . . . 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 1119	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2o) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
13	9, 12	biimtrrid 242	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 1o ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
14	13	expd 414	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1o ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))))
15	14	3impia 1115	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))))
16		oe1m 8547	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)
17	16	adantr 479	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = 1o)
18		oe1m 8547	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o)
19	18	adantl 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐵) = 1o)
20	17, 19	eqtr4d 2773	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ↑o 𝐴) = (1o ↑o 𝐵))
21		eqimss 4039	. . . . . . . 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 7418	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐴) = (𝐶 ↑o 𝐴))
24		oveq1 7418	. . . . . . . 8 ⊢ (1o = 𝐶 → (1o ↑o 𝐵) = (𝐶 ↑o 𝐵))
25	23, 24	sseq12d 4014	. . . . . . 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 1130	. . . . 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 855	. . 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 405	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ∖ cdif 3944   ⊆ wss 3947  ∅c0 4321  Ord word 6362  Oncon0 6363  (class class class)co 7411  1_oc1o 8461  2_oc2o 8462   ↑_o coe 8467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-omul 8473  df-oexp 8474

This theorem is referenced by:  oelim2  8597  oeoalem  8598  oeoelem  8600  oaabs2  8650  cantnflt  9669  cnfcom  9697  oege1  42358  cantnf2  42377  omabs2  42384  omltoe  42460