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Theorem oewordi 8542
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 6331 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
2 ordgt0ge1 8443 . . . . . 6 (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1o𝐶))
31, 2syl 17 . . . . 5 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1o𝐶))
4 1on 8428 . . . . . 6 1o ∈ On
5 onsseleq 6362 . . . . . 6 ((1o ∈ On ∧ 𝐶 ∈ On) → (1o𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
64, 5mpan 689 . . . . 5 (𝐶 ∈ On → (1o𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
73, 6bitrd 279 . . . 4 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
873ad2ant3 1136 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1o𝐶 ∨ 1o = 𝐶)))
9 ondif2 8452 . . . . . . 7 (𝐶 ∈ (On ∖ 2o) ↔ (𝐶 ∈ On ∧ 1o𝐶))
10 oeword 8541 . . . . . . . . 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 1122 . . . . . . 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 417 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (1o𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵)))))
15143impia 1118 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1o𝐶 → (𝐴𝐵 → (𝐶o 𝐴) ⊆ (𝐶o 𝐵))))
16 oe1m 8496 . . . . . . . . . 10 (𝐴 ∈ On → (1oo 𝐴) = 1o)
1716adantr 482 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) = 1o)
18 oe1m 8496 . . . . . . . . . 10 (𝐵 ∈ On → (1oo 𝐵) = 1o)
1918adantl 483 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐵) = 1o)
2017, 19eqtr4d 2776 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) = (1oo 𝐵))
21 eqimss 4004 . . . . . . . 8 ((1oo 𝐴) = (1oo 𝐵) → (1oo 𝐴) ⊆ (1oo 𝐵))
2220, 21syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1oo 𝐴) ⊆ (1oo 𝐵))
23 oveq1 7368 . . . . . . . 8 (1o = 𝐶 → (1oo 𝐴) = (𝐶o 𝐴))
24 oveq1 7368 . . . . . . . 8 (1o = 𝐶 → (1oo 𝐵) = (𝐶o 𝐵))
2523, 24sseq12d 3981 . . . . . . 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 1133 . . . . 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 858 . . 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 408 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  cdif 3911  wss 3914  c0 4286  Ord word 6320  Oncon0 6321  (class class class)co 7361  1oc1o 8409  2oc2o 8410  o coe 8415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-omul 8421  df-oexp 8422
This theorem is referenced by:  oelim2  8546  oeoalem  8547  oeoelem  8549  oaabs2  8599  cantnflt  9616  cnfcom  9644  oege1  41688  cantnf2  41707  omabs2  41714  omltoe  41771
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