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Theorem oewordi 8627
Description: Weak ordering property of ordinal exponentiation. Lemma 3.19 of [Schloeder] p. 10. (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 6395 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
2 ordgt0ge1 8529 . . . . . 6 (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o𝐶))
31, 2syl 17 . . . . 5 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o𝐶))
4 1on 8516 . . . . . 6 1o ∈ On
5 onsseleq 6426 . . . . . 6 ((1o ∈ On ∧ 𝐶 ∈ On) → (1o𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
64, 5mpan 690 . . . . 5 (𝐶 ∈ On → (1o𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
73, 6bitrd 279 . . . 4 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
873ad2ant3 1134 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
9 ondif2 8538 . . . . . . 7 (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o𝐶))
10 oeword 8626 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
1110biimpd 229 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
12113expia 1120 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2o) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
139, 12biimtrrid 243 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐶 ∈ On ∧ 1o𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
1413expd 415 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1o𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))))
15143impia 1116 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
16 oe1m 8581 . . . . . . . . . 10 (𝐴 ∈ On → (1oo 𝐴) = 1o)
1716adantr 480 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) = 1o)
18 oe1m 8581 . . . . . . . . . 10 (𝐵 ∈ On → (1oo 𝐵) = 1o)
1918adantl 481 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐵) = 1o)
2017, 19eqtr4d 2777 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) = (1oo 𝐵))
21 eqimss 4053 . . . . . . . 8 ((1oo 𝐴) = (1oo 𝐵) → (1oo 𝐴) ⊆ (1oo 𝐵))
2220, 21syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) ⊆ (1oo 𝐵))
23 oveq1 7437 . . . . . . . 8 (1o = 𝐶 → (1oo 𝐴) = (𝐶o 𝐴))
24 oveq1 7437 . . . . . . . 8 (1o = 𝐶 → (1oo 𝐵) = (𝐶o 𝐵))
2523, 24sseq12d 4028 . . . . . . 7 (1o = 𝐶 → ((1oo 𝐴) ⊆ (1oo 𝐵) ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
2622, 25syl5ibcom 245 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o = 𝐶 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
27263adant3 1131 . . . . 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 859 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((1o𝐶 ∨ 1o = 𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
308, 29sylbid 240 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
3130imp 406 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  cdif 3959  wss 3962  c0 4338  Ord word 6384  Oncon0 6385  (class class class)co 7430  1oc1o 8497  2oc2o 8498  o coe 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-oexp 8510
This theorem is referenced by:  oelim2  8631  oeoalem  8632  oeoelem  8634  oaabs2  8685  cantnflt  9709  cnfcom  9737  oege1  43295  cantnf2  43314  omabs2  43321  omltoe  43396
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