Theorem oewordi 7671

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 5733	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordgt0ge1 7577	. . . . . 6 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1𝑜 ⊆ 𝐶))
3	1, 2	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1𝑜 ⊆ 𝐶))
4		1on 7567	. . . . . 6 ⊢ 1𝑜 ∈ On
5		onsseleq 5765	. . . . . 6 ⊢ ((1𝑜 ∈ On ∧ 𝐶 ∈ On) → (1𝑜 ⊆ 𝐶 ↔ (1𝑜 ∈ 𝐶 ∨ 1𝑜 = 𝐶)))
6	4, 5	mpan 706	. . . . 5 ⊢ (𝐶 ∈ On → (1𝑜 ⊆ 𝐶 ↔ (1𝑜 ∈ 𝐶 ∨ 1𝑜 = 𝐶)))
7	3, 6	bitrd 268	. . . 4 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1𝑜 ∈ 𝐶 ∨ 1𝑜 = 𝐶)))
8	7	3ad2ant3 1084	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1𝑜 ∈ 𝐶 ∨ 1𝑜 = 𝐶)))
9		ondif2 7582	. . . . . . 7 ⊢ (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜 ∈ 𝐶))
10		oeword 7670	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))
11	10	biimpd 219	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))
12	11	3expia 1267	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐴 ⊆ 𝐵 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵))))
13	9, 12	syl5bir 233	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 1𝑜 ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵))))
14	13	expd 452	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1𝑜 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))))
15	14	3impia 1261	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1𝑜 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵))))
16		oe1m 7625	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (1𝑜 ↑𝑜 𝐴) = 1𝑜)
17	16	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 ↑𝑜 𝐴) = 1𝑜)
18		oe1m 7625	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (1𝑜 ↑𝑜 𝐵) = 1𝑜)
19	18	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 ↑𝑜 𝐵) = 1𝑜)
20	17, 19	eqtr4d 2659	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 ↑𝑜 𝐴) = (1𝑜 ↑𝑜 𝐵))
21		eqimss 3657	. . . . . . . 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 6657	. . . . . . . 8 ⊢ (1𝑜 = 𝐶 → (1𝑜 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐴))
24		oveq1 6657	. . . . . . . 8 ⊢ (1𝑜 = 𝐶 → (1𝑜 ↑𝑜 𝐵) = (𝐶 ↑𝑜 𝐵))
25	23, 24	sseq12d 3634	. . . . . . 7 ⊢ (1𝑜 = 𝐶 → ((1𝑜 ↑𝑜 𝐴) ⊆ (1𝑜 ↑𝑜 𝐵) ↔ (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))
26	22, 25	syl5ibcom 235	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 = 𝐶 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))
27	26	3adant3 1081	. . . . 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 395	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((1𝑜 ∈ 𝐶 ∨ 1𝑜 = 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵))))
30	8, 29	sylbid 230	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵))))
31	30	imp 445	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  Ord word 5722  Oncon0 5723  (class class class)co 6650  1_𝑜c1o 7553  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:  oelim2  7675  oeoalem  7676  oeoelem  7678  oaabs2  7725  cantnflt  8569  cnfcom  8597