Theorem oeord 8515

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 8514	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
2	1	3adant1 1131	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
3		oveq2 7366	. . . . . 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 8514	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐴 → (𝐶 ↑o 𝐵) ∈ (𝐶 ↑o 𝐴)))
6	5	3adant2 1132	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐴 → (𝐶 ↑o 𝐵) ∈ (𝐶 ↑o 𝐴)))
7	4, 6	orim12d 967	. . . 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 4072	. . . . . 6 ⊢ (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
10	9	3ad2ant3 1136	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
11		simp1 1137	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐴 ∈ On)
12		oecl 8463	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ On)
13	10, 11, 12	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶 ↑o 𝐴) ∈ On)
14		simp2 1138	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐵 ∈ On)
15		oecl 8463	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑o 𝐵) ∈ On)
16	10, 14, 15	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐶 ↑o 𝐵) ∈ On)
17		eloni 6325	. . . . 5 ⊢ ((𝐶 ↑o 𝐴) ∈ On → Ord (𝐶 ↑o 𝐴))
18		eloni 6325	. . . . 5 ⊢ ((𝐶 ↑o 𝐵) ∈ On → Ord (𝐶 ↑o 𝐵))
19		ordtri2 6350	. . . . 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 597	. . . 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 585	. . 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 6325	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
23		eloni 6325	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		ordtri2 6350	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
25	22, 23, 24	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
26	25	3adant3 1133	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
27	8, 21, 26	3imtr4d 294	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵) → 𝐴 ∈ 𝐵))
28	2, 27	impbid 212	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887  Ord word 6314  Oncon0 6315  (class class class)co 7358  2_oc2o 8390   ↑_o coe 8395

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-omul 8401  df-oexp 8402

This theorem is referenced by:  oeword  8517  oeeui  8529  omabs  8578  cantnflem3  9601  oeord2com  43742  omabs2  43763