Theorem oeord 8555

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 8554	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
2	1	3adant1 1130	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
3		oveq2 7385	. . . . . 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 8554	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐴 → (𝐶 ↑o 𝐵) ∈ (𝐶 ↑o 𝐴)))
6	5	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐴 → (𝐶 ↑o 𝐵) ∈ (𝐶 ↑o 𝐴)))
7	4, 6	orim12d 963	. . . 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 4106	. . . . . 6 ⊢ (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
10	9	3ad2ant3 1135	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
11		simp1 1136	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐴 ∈ On)
12		oecl 8503	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ On)
13	10, 11, 12	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶 ↑o 𝐴) ∈ On)
14		simp2 1137	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐵 ∈ On)
15		oecl 8503	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑o 𝐵) ∈ On)
16	10, 14, 15	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶 ↑o 𝐵) ∈ On)
17		eloni 6347	. . . . 5 ⊢ ((𝐶 ↑o 𝐴) ∈ On → Ord (𝐶 ↑o 𝐴))
18		eloni 6347	. . . . 5 ⊢ ((𝐶 ↑o 𝐵) ∈ On → Ord (𝐶 ↑o 𝐵))
19		ordtri2 6372	. . . . 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 596	. . . 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 584	. . 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 6347	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
23		eloni 6347	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		ordtri2 6372	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
25	22, 23, 24	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
26	25	3adant3 1132	. . 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 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∖ cdif 3925  Ord word 6336  Oncon0 6337  (class class class)co 7377  2_oc2o 8426   ↑_o coe 8431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pr 5404  ax-un 7692

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-omul 8437  df-oexp 8438

This theorem is referenced by:  oeword  8557  oeeui  8569  omabs  8617  cantnflem3  9651  oeord2com  41737  omabs2  41758