Theorem oecan 8585

Description: Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Assertion

Ref	Expression
oecan	⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ 𝐵 = 𝐶))

Proof of Theorem oecan

Step	Hyp	Ref	Expression
1		oeordi 8583	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
2	1	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
3	2	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
4		oeordi 8583	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐶 ∈ 𝐵 → (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)))
5	4	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → (𝐶 ∈ 𝐵 → (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)))
6	5	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ∈ 𝐵 → (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)))
7	3, 6	orim12d 963	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵) → ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵))))
8	7	con3d 152	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)) → ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
9		eldifi 4125	. . . . . 6 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
10	9	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11		simp2 1137	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
12		oecl 8533	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
13	10, 11, 12	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
14		simp3 1138	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
15		oecl 8533	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
16	10, 14, 15	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
17		eloni 6371	. . . . 5 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵))
18		eloni 6371	. . . . 5 ⊢ ((𝐴 ↑o 𝐶) ∈ On → Ord (𝐴 ↑o 𝐶))
19		ordtri3 6397	. . . . 5 ⊢ ((Ord (𝐴 ↑o 𝐵) ∧ Ord (𝐴 ↑o 𝐶)) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ ¬ ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵))))
20	17, 18, 19	syl2an 596	. . . 4 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ (𝐴 ↑o 𝐶) ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ ¬ ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵))))
21	13, 16, 20	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ ¬ ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵))))
22		eloni 6371	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
23		eloni 6371	. . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶)
24		ordtri3 6397	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
25	22, 23, 24	syl2an 596	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
26	25	3adant1 1130	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
27	8, 21, 26	3imtr4d 293	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) → 𝐵 = 𝐶))
28		oveq2 7413	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶))
29	27, 28	impbid1 224	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∖ cdif 3944  Ord word 6360  Oncon0 6361  (class class class)co 7405  2_oc2o 8456   ↑_o coe 8461

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-oexp 8468

This theorem is referenced by:  oeword  8586  infxpenc2lem1  10010