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Theorem oewordi 7616
 Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)
Assertion
Ref Expression
oewordi (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))

Proof of Theorem oewordi
StepHypRef Expression
1 eloni 5692 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
2 ordgt0ge1 7522 . . . . . 6 (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1𝑜𝐶))
31, 2syl 17 . . . . 5 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1𝑜𝐶))
4 1on 7512 . . . . . 6 1𝑜 ∈ On
5 onsseleq 5724 . . . . . 6 ((1𝑜 ∈ On ∧ 𝐶 ∈ On) → (1𝑜𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
64, 5mpan 705 . . . . 5 (𝐶 ∈ On → (1𝑜𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
73, 6bitrd 268 . . . 4 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
873ad2ant3 1082 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
9 ondif2 7527 . . . . . . 7 (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜𝐶))
10 oeword 7615 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 ↔ (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))
1110biimpd 219 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2𝑜)) → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))
12113expia 1264 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2𝑜) → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵))))
139, 12syl5bir 233 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 1𝑜𝐶) → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵))))
1413expd 452 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1𝑜𝐶 → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))))
15143impia 1258 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1𝑜𝐶 → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵))))
16 oe1m 7570 . . . . . . . . . 10 (𝐴 ∈ On → (1𝑜𝑜 𝐴) = 1𝑜)
1716adantr 481 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) = 1𝑜)
18 oe1m 7570 . . . . . . . . . 10 (𝐵 ∈ On → (1𝑜𝑜 𝐵) = 1𝑜)
1918adantl 482 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐵) = 1𝑜)
2017, 19eqtr4d 2658 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) = (1𝑜𝑜 𝐵))
21 eqimss 3636 . . . . . . . 8 ((1𝑜𝑜 𝐴) = (1𝑜𝑜 𝐵) → (1𝑜𝑜 𝐴) ⊆ (1𝑜𝑜 𝐵))
2220, 21syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) ⊆ (1𝑜𝑜 𝐵))
23 oveq1 6611 . . . . . . . 8 (1𝑜 = 𝐶 → (1𝑜𝑜 𝐴) = (𝐶𝑜 𝐴))
24 oveq1 6611 . . . . . . . 8 (1𝑜 = 𝐶 → (1𝑜𝑜 𝐵) = (𝐶𝑜 𝐵))
2523, 24sseq12d 3613 . . . . . . 7 (1𝑜 = 𝐶 → ((1𝑜𝑜 𝐴) ⊆ (1𝑜𝑜 𝐵) ↔ (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))
2622, 25syl5ibcom 235 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜 = 𝐶 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))
27263adant3 1079 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1𝑜 = 𝐶 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))
2827a1dd 50 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1𝑜 = 𝐶 → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵))))
2915, 28jaod 395 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((1𝑜𝐶 ∨ 1𝑜 = 𝐶) → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵))))
308, 29sylbid 230 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵))))
3130imp 445 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ∖ cdif 3552   ⊆ wss 3555  ∅c0 3891  Ord word 5681  Oncon0 5682  (class class class)co 6604  1𝑜c1o 7498  2𝑜c2o 7499   ↑𝑜 coe 7504 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511 This theorem is referenced by:  oelim2  7620  oeoalem  7621  oeoelem  7623  oaabs2  7670  cantnflt  8513  cnfcom  8541
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