Theorem oecan 8512

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 8510	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
2	1	ancoms 458	. . . . . 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 8510	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝐶 ∈ 𝐵 → (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)))
5	4	ancoms 458	. . . . . 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 966	. . . 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 4080	. . . . . 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 8460	. . . . 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 8460	. . . . 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 6323	. . . . 5 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵))
18		eloni 6323	. . . . 5 ⊢ ((𝐴 ↑o 𝐶) ∈ On → Ord (𝐴 ↑o 𝐶))
19		ordtri3 6349	. . . . 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 6323	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
23		eloni 6323	. . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶)
24		ordtri3 6349	. . . . 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 294	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) → 𝐵 = 𝐶))
28		oveq2 7362	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶))
29	27, 28	impbid1 225	1 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∖ cdif 3895  Ord word 6312  Oncon0 6313  (class class class)co 7354  2_oc2o 8387   ↑_o coe 8392

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7676

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7805  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-1o 8393  df-2o 8394  df-oadd 8397  df-omul 8398  df-oexp 8399

This theorem is referenced by:  oeword  8513  infxpenc2lem1  9919