Theorem oecan 7556

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 7554	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐵 ∈ 𝐶 → (𝐴 ↑𝑜 𝐵) ∈ (𝐴 ↑𝑜 𝐶)))
2	1	ancoms 468	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ↑𝑜 𝐵) ∈ (𝐴 ↑𝑜 𝐶)))
3	2	3adant2 1073	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ↑𝑜 𝐵) ∈ (𝐴 ↑𝑜 𝐶)))
4		oeordi 7554	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2𝑜)) → (𝐶 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ∈ (𝐴 ↑𝑜 𝐵)))
5	4	ancoms 468	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On) → (𝐶 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ∈ (𝐴 ↑𝑜 𝐵)))
6	5	3adant3 1074	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ∈ 𝐵 → (𝐴 ↑𝑜 𝐶) ∈ (𝐴 ↑𝑜 𝐵)))
7	3, 6	orim12d 879	. . . 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 147	. . 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 3694	. . . . . 6 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
10	9	3ad2ant1 1075	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11		simp2 1055	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
12		oecl 7504	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
13	10, 11, 12	syl2anc 691	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
14		simp3 1056	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
15		oecl 7504	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 𝐶) ∈ On)
16	10, 14, 15	syl2anc 691	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 𝐶) ∈ On)
17		eloni 5650	. . . . 5 ⊢ ((𝐴 ↑𝑜 𝐵) ∈ On → Ord (𝐴 ↑𝑜 𝐵))
18		eloni 5650	. . . . 5 ⊢ ((𝐴 ↑𝑜 𝐶) ∈ On → Ord (𝐴 ↑𝑜 𝐶))
19		ordtri3 5676	. . . . 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 493	. . . 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 691	. . 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 5650	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
23		eloni 5650	. . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶)
24		ordtri3 5676	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
25	22, 23, 24	syl2an 493	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
26	25	3adant1 1072	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
27	8, 21, 26	3imtr4d 282	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) = (𝐴 ↑𝑜 𝐶) → 𝐵 = 𝐶))
28		oveq2 6557	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ↑𝑜 𝐵) = (𝐴 ↑𝑜 𝐶))
29	27, 28	impbid1 214	1 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑𝑜 𝐵) = (𝐴 ↑𝑜 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  Ord word 5639  Oncon0 5640  (class class class)co 6549  2_𝑜c2o 7441   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oeword  7557  infxpenc2lem1  8725