Theorem oeword 7557

Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)

Assertion

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

Proof of Theorem oeword

Step	Hyp	Ref	Expression
1		oeord 7555	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵)))
2		oecan 7556	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ↔ 𝐴 = 𝐵))
3	2	3coml 1264	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵) ↔ 𝐴 = 𝐵))
4	3	bicomd 212	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 = 𝐵 ↔ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵)))
5	1, 4	orbi12d 742	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵))))
6		onsseleq 5682	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	6	3adant3 1074	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8		eldifi 3694	. . . 4 ⊢ (𝐶 ∈ (On ∖ 2𝑜) → 𝐶 ∈ On)
9		id 22	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10		oecl 7504	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑𝑜 𝐴) ∈ On)
11		oecl 7504	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑𝑜 𝐵) ∈ On)
12	10, 11	anim12dan 878	. . . . 5 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑𝑜 𝐴) ∈ On ∧ (𝐶 ↑𝑜 𝐵) ∈ On))
13		onsseleq 5682	. . . . 5 ⊢ (((𝐶 ↑𝑜 𝐴) ∈ On ∧ (𝐶 ↑𝑜 𝐵) ∈ On) → ((𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵))))
14	12, 13	syl 17	. . . 4 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵))))
15	8, 9, 14	syl2anr 494	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵))))
16	15	3impa 1251	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → ((𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵) ↔ ((𝐶 ↑𝑜 𝐴) ∈ (𝐶 ↑𝑜 𝐵) ∨ (𝐶 ↑𝑜 𝐴) = (𝐶 ↑𝑜 𝐵))))
17	5, 7, 16	3bitr4d 299	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑𝑜 𝐴) ⊆ (𝐶 ↑𝑜 𝐵)))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  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:  oewordi  7558