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Theorem oewordi 8590
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 6374 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
2 ordgt0ge1 8492 . . . . . 6 (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o𝐶))
31, 2syl 17 . . . . 5 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o𝐶))
4 1on 8477 . . . . . 6 1o ∈ On
5 onsseleq 6405 . . . . . 6 ((1o ∈ On ∧ 𝐶 ∈ On) → (1o𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
64, 5mpan 688 . . . . 5 (𝐶 ∈ On → (1o𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
73, 6bitrd 278 . . . 4 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
873ad2ant3 1135 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
9 ondif2 8501 . . . . . . 7 (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o𝐶))
10 oeword 8589 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
1110biimpd 228 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
12113expia 1121 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (On ∖ 2o) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
139, 12biimtrrid 242 . . . . . 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 1117 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
16 oe1m 8544 . . . . . . . . . 10 (𝐴 ∈ On → (1oo 𝐴) = 1o)
1716adantr 481 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) = 1o)
18 oe1m 8544 . . . . . . . . . 10 (𝐵 ∈ On → (1oo 𝐵) = 1o)
1918adantl 482 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐵) = 1o)
2017, 19eqtr4d 2775 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) = (1oo 𝐵))
21 eqimss 4040 . . . . . . . 8 ((1oo 𝐴) = (1oo 𝐵) → (1oo 𝐴) ⊆ (1oo 𝐵))
2220, 21syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) ⊆ (1oo 𝐵))
23 oveq1 7415 . . . . . . . 8 (1o = 𝐶 → (1oo 𝐴) = (𝐶o 𝐴))
24 oveq1 7415 . . . . . . . 8 (1o = 𝐶 → (1oo 𝐵) = (𝐶o 𝐵))
2523, 24sseq12d 4015 . . . . . . 7 (1o = 𝐶 → ((1oo 𝐴) ⊆ (1oo 𝐵) ↔ (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
2622, 25syl5ibcom 244 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o = 𝐶 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))
27263adant3 1132 . . . . 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 857 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((1o𝐶 ∨ 1o = 𝐶) → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
308, 29sylbid 239 . 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  cdif 3945  wss 3948  c0 4322  Ord word 6363  Oncon0 6364  (class class class)co 7408  1oc1o 8458  2oc2o 8459  o coe 8464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-2o 8466  df-oadd 8469  df-omul 8470  df-oexp 8471
This theorem is referenced by:  oelim2  8594  oeoalem  8595  oeoelem  8597  oaabs2  8647  cantnflt  9666  cnfcom  9694  oege1  42046  cantnf2  42065  omabs2  42072  omltoe  42148
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