Theorem oeord 8064

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 8063	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
2	1	3adant1 1123	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
3		oveq2 7024	. . . . . 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 8063	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐴 → (𝐶 ↑o 𝐵) ∈ (𝐶 ↑o 𝐴)))
6	5	3adant2 1124	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐴 → (𝐶 ↑o 𝐵) ∈ (𝐶 ↑o 𝐴)))
7	4, 6	orim12d 959	. . . 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 155	. . 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 4024	. . . . . 6 ⊢ (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
10	9	3ad2ant3 1128	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐶 ∈ On)
11		simp1 1129	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐴 ∈ On)
12		oecl 8013	. . . . 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 1130	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → 𝐵 ∈ On)
15		oecl 8013	. . . . 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 6076	. . . . 5 ⊢ ((𝐶 ↑o 𝐴) ∈ On → Ord (𝐶 ↑o 𝐴))
18		eloni 6076	. . . . 5 ⊢ ((𝐶 ↑o 𝐵) ∈ On → Ord (𝐶 ↑o 𝐵))
19		ordtri2 6101	. . . . 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 595	. . . 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 6076	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
23		eloni 6076	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		ordtri2 6101	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
25	22, 23, 24	syl2an 595	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
26	25	3adant3 1125	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
27	8, 21, 26	3imtr4d 295	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵) → 𝐴 ∈ 𝐵))
28	2, 27	impbid 213	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∨ wo 842   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ∖ cdif 3856  Ord word 6065  Oncon0 6066  (class class class)co 7016  2_oc2o 7947   ↑_o coe 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-omul 7958  df-oexp 7959

This theorem is referenced by:  oeword  8066  oeeui  8078  omabs  8124  cantnflem3  9000