Theorem oecan 8645

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 8643	. . . . . . 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 8643	. . . . . . 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 965	. . . 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 4154	. . . . . 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 8593	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
13	10, 11, 12	syl2anc 583	. . . 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 8593	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
16	10, 14, 15	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On)
17		eloni 6405	. . . . 5 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵))
18		eloni 6405	. . . . 5 ⊢ ((𝐴 ↑o 𝐶) ∈ On → Ord (𝐴 ↑o 𝐶))
19		ordtri3 6431	. . . . 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 595	. . . 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 583	. . 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 6405	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
23		eloni 6405	. . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶)
24		ordtri3 6431	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
25	22, 23, 24	syl2an 595	. . . 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 7456	. 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 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973  Ord word 6394  Oncon0 6395  (class class class)co 7448  2_oc2o 8516   ↑_o coe 8521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-omul 8527  df-oexp 8528

This theorem is referenced by:  oeword  8646  infxpenc2lem1  10088