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Theorem oewordi 7716
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 5771 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
2 ordgt0ge1 7622 . . . . . 6 (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1𝑜𝐶))
31, 2syl 17 . . . . 5 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1𝑜𝐶))
4 1on 7612 . . . . . 6 1𝑜 ∈ On
5 onsseleq 5803 . . . . . 6 ((1𝑜 ∈ On ∧ 𝐶 ∈ On) → (1𝑜𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
64, 5mpan 706 . . . . 5 (𝐶 ∈ On → (1𝑜𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
73, 6bitrd 268 . . . 4 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
873ad2ant3 1104 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
9 ondif2 7627 . . . . . . 7 (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜𝐶))
10 oeword 7715 . . . . . . . . 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 1286 . . . . . . 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 451 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1𝑜𝐶 → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵)))))
15143impia 1280 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1𝑜𝐶 → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵))))
16 oe1m 7670 . . . . . . . . . 10 (𝐴 ∈ On → (1𝑜𝑜 𝐴) = 1𝑜)
1716adantr 480 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) = 1𝑜)
18 oe1m 7670 . . . . . . . . . 10 (𝐵 ∈ On → (1𝑜𝑜 𝐵) = 1𝑜)
1918adantl 481 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐵) = 1𝑜)
2017, 19eqtr4d 2688 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) = (1𝑜𝑜 𝐵))
21 eqimss 3690 . . . . . . . 8 ((1𝑜𝑜 𝐴) = (1𝑜𝑜 𝐵) → (1𝑜𝑜 𝐴) ⊆ (1𝑜𝑜 𝐵))
2220, 21syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) ⊆ (1𝑜𝑜 𝐵))
23 oveq1 6697 . . . . . . . 8 (1𝑜 = 𝐶 → (1𝑜𝑜 𝐴) = (𝐶𝑜 𝐴))
24 oveq1 6697 . . . . . . . 8 (1𝑜 = 𝐶 → (1𝑜𝑜 𝐵) = (𝐶𝑜 𝐵))
2523, 24sseq12d 3667 . . . . . . 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 1101 . . . . 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 394 . . 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 444 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 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  cdif 3604  wss 3607  c0 3948  Ord word 5760  Oncon0 5761  (class class class)co 6690  1𝑜c1o 7598  2𝑜c2o 7599  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by:  oelim2  7720  oeoalem  7721  oeoelem  7723  oaabs2  7770  cantnflt  8607  cnfcom  8635
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