Theorem oeord 8592

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 8591	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
2	1	3adant1 1128	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
3		oveq2 7421	. . . . . 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 8591	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐴 → (𝐶 ↑o 𝐵) ∈ (𝐶 ↑o 𝐴)))
6	5	3adant2 1129	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐴 → (𝐶 ↑o 𝐵) ∈ (𝐶 ↑o 𝐴)))
7	4, 6	orim12d 961	. . . 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 152	. . 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 4127	. . . . . 6 ⊢ (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
10	9	3ad2ant3 1133	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
11		simp1 1134	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐴 ∈ On)
12		oecl 8541	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ On)
13	10, 11, 12	syl2anc 582	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶 ↑o 𝐴) ∈ On)
14		simp2 1135	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐵 ∈ On)
15		oecl 8541	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑o 𝐵) ∈ On)
16	10, 14, 15	syl2anc 582	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶 ↑o 𝐵) ∈ On)
17		eloni 6375	. . . . 5 ⊢ ((𝐶 ↑o 𝐴) ∈ On → Ord (𝐶 ↑o 𝐴))
18		eloni 6375	. . . . 5 ⊢ ((𝐶 ↑o 𝐵) ∈ On → Ord (𝐶 ↑o 𝐵))
19		ordtri2 6400	. . . . 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 594	. . . 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 582	. . 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 6375	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
23		eloni 6375	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		ordtri2 6400	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
25	22, 23, 24	syl2an 594	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
26	25	3adant3 1130	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
27	8, 21, 26	3imtr4d 293	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵) → 𝐴 ∈ 𝐵))
28	2, 27	impbid 211	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ∖ cdif 3946  Ord word 6364  Oncon0 6365  (class class class)co 7413  2_oc2o 8464   ↑_o coe 8469

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 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729

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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-omul 8475  df-oexp 8476

This theorem is referenced by:  oeword  8594  oeeui  8606  omabs  8654  cantnflem3  9690  oeord2com  42365  omabs2  42386