Theorem oeord 8509

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 8508	. . 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 7360	. . . . . 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 8508	. . . . . 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 966	. . . 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 4080	. . . . . 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 8458	. . . . 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 8458	. . . . 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 6322	. . . . 5 ⊢ ((𝐶 ↑o 𝐴) ∈ On → Ord (𝐶 ↑o 𝐴))
18		eloni 6322	. . . . 5 ⊢ ((𝐶 ↑o 𝐵) ∈ On → Ord (𝐶 ↑o 𝐵))
19		ordtri2 6347	. . . . 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 6322	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
23		eloni 6322	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		ordtri2 6347	. . . . 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 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 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∖ cdif 3894  Ord word 6311  Oncon0 6312  (class class class)co 7352  2_oc2o 8385   ↑_o coe 8390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-oadd 8395  df-omul 8396  df-oexp 8397

This theorem is referenced by:  oeword  8511  oeeui  8523  omabs  8572  cantnflem3  9587  oeord2com  43409  omabs2  43430