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Theorem oewordi 8206
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 6194 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
2 ordgt0ge1 8111 . . . . . 6 (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o𝐶))
31, 2syl 17 . . . . 5 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o𝐶))
4 1on 8098 . . . . . 6 1o ∈ On
5 onsseleq 6225 . . . . . 6 ((1o ∈ On ∧ 𝐶 ∈ On) → (1o𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
64, 5mpan 686 . . . . 5 (𝐶 ∈ On → (1o𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
73, 6bitrd 280 . . . 4 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
873ad2ant3 1127 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
9 ondif2 8116 . . . . . . 7 (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o𝐶))
10 oeword 8205 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
1110biimpd 230 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
12113expia 1113 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2o) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
139, 12syl5bir 244 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 1o𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
1413expd 416 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1o𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))))
15143impia 1109 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
16 oe1m 8160 . . . . . . . . . 10 (𝐴 ∈ On → (1oo 𝐴) = 1o)
1716adantr 481 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) = 1o)
18 oe1m 8160 . . . . . . . . . 10 (𝐵 ∈ On → (1oo 𝐵) = 1o)
1918adantl 482 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐵) = 1o)
2017, 19eqtr4d 2856 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) = (1oo 𝐵))
21 eqimss 4020 . . . . . . . 8 ((1oo 𝐴) = (1oo 𝐵) → (1oo 𝐴) ⊆ (1oo 𝐵))
2220, 21syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) ⊆ (1oo 𝐵))
23 oveq1 7152 . . . . . . . 8 (1o = 𝐶 → (1oo 𝐴) = (𝐶o 𝐴))
24 oveq1 7152 . . . . . . . 8 (1o = 𝐶 → (1oo 𝐵) = (𝐶o 𝐵))
2523, 24sseq12d 3997 . . . . . . 7 (1o = 𝐶 → ((1oo 𝐴) ⊆ (1oo 𝐵) ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
2622, 25syl5ibcom 246 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o = 𝐶 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
27263adant3 1124 . . . . 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 853 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((1o𝐶 ∨ 1o = 𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
308, 29sylbid 241 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
3130imp 407 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  cdif 3930  wss 3933  c0 4288  Ord word 6183  Oncon0 6184  (class class class)co 7145  1oc1o 8084  2oc2o 8085  o coe 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-oexp 8097
This theorem is referenced by:  oelim2  8210  oeoalem  8211  oeoelem  8213  oaabs2  8261  cantnflt  9123  cnfcom  9151
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