Theorem oeword 8592

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 8590	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵)))
2		oecan 8591	. . . . 5 ⊢ ((𝐶 ∈ (On ∖ 2o) ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵) ↔ 𝐴 = 𝐵))
3	2	3coml 1125	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → ((𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵) ↔ 𝐴 = 𝐵))
4	3	bicomd 222	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 = 𝐵 ↔ (𝐶 ↑o 𝐴) = (𝐶 ↑o 𝐵)))
5	1, 4	orbi12d 915	. 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 6404	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	6	3adant3 1130	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8		eldifi 4125	. . . 4 ⊢ (𝐶 ∈ (On ∖ 2o) → 𝐶 ∈ On)
9		id 22	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
10		oecl 8539	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ↑o 𝐴) ∈ On)
11		oecl 8539	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ↑o 𝐵) ∈ On)
12	10, 11	anim12dan 617	. . . . 5 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ↑o 𝐴) ∈ On ∧ (𝐶 ↑o 𝐵) ∈ On))
13		onsseleq 6404	. . . . 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 595	. . 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 1108	. 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 310	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ∖ cdif 3944   ⊆ wss 3947  Oncon0 6363  (class class class)co 7411  2_oc2o 8462   ↑_o coe 8467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-omul 8473  df-oexp 8474

This theorem is referenced by:  oewordi  8593  cantnftermord  42372  omabs2  42384