Theorem oecan 8517

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 8515	. . . . . . 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 8515	. . . . . . 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 4083	. . . . . 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 8464	. . . . 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 8464	. . . . 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 6327	. . . . 5 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵))
18		eloni 6327	. . . . 5 ⊢ ((𝐴 ↑o 𝐶) ∈ On → Ord (𝐴 ↑o 𝐶))
19		ordtri3 6353	. . . . 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 6327	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
23		eloni 6327	. . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶)
24		ordtri3 6353	. . . . 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 7366	. 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 3898  Ord word 6316  Oncon0 6317  (class class class)co 7358  2_oc2o 8391   ↑_o coe 8396

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-omul 8402  df-oexp 8403

This theorem is referenced by:  oeword  8518  infxpenc2lem1  9929