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Theorem oewordi 7838
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 5892 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
2 ordgt0ge1 7744 . . . . . 6 (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 1𝑜𝐶))
31, 2syl 17 . . . . 5 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 1𝑜𝐶))
4 1on 7734 . . . . . 6 1𝑜 ∈ On
5 onsseleq 5924 . . . . . 6 ((1𝑜 ∈ On ∧ 𝐶 ∈ On) → (1𝑜𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
64, 5mpan 708 . . . . 5 (𝐶 ∈ On → (1𝑜𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
73, 6bitrd 268 . . . 4 (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
873ad2ant3 1130 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ (1𝑜𝐶 ∨ 1𝑜 = 𝐶)))
9 ondif2 7749 . . . . . . 7 (𝐶 ∈ (On ∖ 2𝑜) ↔ (𝐶 ∈ On ∧ 1𝑜𝐶))
10 oeword 7837 . . . . . . . . 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 1115 . . . . . . 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 1110 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (1𝑜𝐶 → (𝐴𝐵 → (𝐶𝑜 𝐴) ⊆ (𝐶𝑜 𝐵))))
16 oe1m 7792 . . . . . . . . . 10 (𝐴 ∈ On → (1𝑜𝑜 𝐴) = 1𝑜)
1716adantr 472 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) = 1𝑜)
18 oe1m 7792 . . . . . . . . . 10 (𝐵 ∈ On → (1𝑜𝑜 𝐵) = 1𝑜)
1918adantl 473 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐵) = 1𝑜)
2017, 19eqtr4d 2795 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) = (1𝑜𝑜 𝐵))
21 eqimss 3796 . . . . . . . 8 ((1𝑜𝑜 𝐴) = (1𝑜𝑜 𝐵) → (1𝑜𝑜 𝐴) ⊆ (1𝑜𝑜 𝐵))
2220, 21syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝑜 𝐴) ⊆ (1𝑜𝑜 𝐵))
23 oveq1 6818 . . . . . . . 8 (1𝑜 = 𝐶 → (1𝑜𝑜 𝐴) = (𝐶𝑜 𝐴))
24 oveq1 6818 . . . . . . . 8 (1𝑜 = 𝐶 → (1𝑜𝑜 𝐵) = (𝐶𝑜 𝐵))
2523, 24sseq12d 3773 . . . . . . 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 1127 . . . . 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 1072   = wceq 1630  wcel 2137  cdif 3710  wss 3713  c0 4056  Ord word 5881  Oncon0 5882  (class class class)co 6811  1𝑜c1o 7720  2𝑜c2o 7721  𝑜 coe 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-omul 7732  df-oexp 7733
This theorem is referenced by:  oelim2  7842  oeoalem  7843  oeoelem  7845  oaabs2  7892  cantnflt  8740  cnfcom  8768
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