Theorem oeord 7668

Description: Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Assertion

Ref	Expression
oeord	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))

Proof of Theorem oeord

Step	Hyp	Ref	Expression
1		oeordi 7667	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
2	1	3adant1 1079	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 → (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
3		oveq2 6658	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵))
4	3	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 = 𝐵 → (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))
5		oeordi 7667	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵 ∈ 𝐴 → (𝐶 ↑𝑜 𝐵) ∈ (𝐶 ↑𝑜 𝐴)))
6	5	3adant2 1080	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐵 ∈ 𝐴 → (𝐶 ↑𝑜 𝐵) ∈ (𝐶 ↑𝑜 𝐴)))
7	4, 6	orim12d 883	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐵) ∈ (𝐶 ↑𝑜 𝐴))))
8	7	con3d 148	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (¬ ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐵) ∈ (𝐶 ↑𝑜 𝐴)) → ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
9		eldifi 3732	. . . . . 6 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
10	9	3ad2ant3 1084	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐶 ∈ On)
11		simp1 1061	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐴 ∈ On)
12		oecl 7617	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On)
13	10, 11, 12	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶 ↑𝑜 𝐴) ∈ On)
14		simp2 1062	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → 𝐵 ∈ On)
15		oecl 7617	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑𝑜 𝐵) ∈ On)
16	10, 14, 15	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐶 ↑𝑜 𝐵) ∈ On)
17		eloni 5733	. . . . 5 ⊢ ((𝐶 ↑𝑜 𝐴) ∈ On → Ord (𝐶 ↑𝑜 𝐴))
18		eloni 5733	. . . . 5 ⊢ ((𝐶 ↑𝑜 𝐵) ∈ On → Ord (𝐶 ↑𝑜 𝐵))
19		ordtri2 5758	. . . . 5 ⊢ ((Ord (𝐶 ↑𝑜 𝐴) ∧ Ord (𝐶 ↑𝑜 𝐵)) → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ↔ ¬ ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐵) ∈ (𝐶 ↑𝑜 𝐴))))
20	17, 18, 19	syl2an 494	. . . 4 ⊢ (((𝐶 ↑𝑜 𝐴) ∈ On ∧ (𝐶 ↑𝑜 𝐵) ∈ On) → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ↔ ¬ ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐵) ∈ (𝐶 ↑𝑜 𝐴))))
21	13, 16, 20	syl2anc 693	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ↔ ¬ ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐵) ∈ (𝐶 ↑𝑜 𝐴))))
22		eloni 5733	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
23		eloni 5733	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		ordtri2 5758	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
25	22, 23, 24	syl2an 494	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
26	25	3adant3 1081	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
27	8, 21, 26	3imtr4d 283	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) → 𝐴 ∈ 𝐵))
28	2, 27	impbid 202	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571  Ord word 5722  Oncon0 5723  (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:  oeword  7670  oeeui  7682  omabs  7727  cantnflem3  8588