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Theorem oeordsuc 8322
Description: Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.)
Assertion
Ref Expression
oeordsuc ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))

Proof of Theorem oeordsuc
StepHypRef Expression
1 onelon 6238 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 416 . . 3 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
32adantr 484 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵𝐴 ∈ On))
4 oewordri 8320 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
543adant1 1132 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
6 oecl 8264 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
763adant2 1133 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
8 oecl 8264 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o 𝐶) ∈ On)
983adant1 1132 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o 𝐶) ∈ On)
10 simp1 1138 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 omwordri 8300 . . . . . . . . . . 11 (((𝐴o 𝐶) ∈ On ∧ (𝐵o 𝐶) ∈ On ∧ 𝐴 ∈ On) → ((𝐴o 𝐶) ⊆ (𝐵o 𝐶) → ((𝐴o 𝐶) ·o 𝐴) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
127, 9, 10, 11syl3anc 1373 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o 𝐶) ⊆ (𝐵o 𝐶) → ((𝐴o 𝐶) ·o 𝐴) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
135, 12syld 47 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴o 𝐶) ·o 𝐴) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
14 oesuc 8254 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
15143adant2 1133 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
1615sseq1d 3932 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ↔ ((𝐴o 𝐶) ·o 𝐴) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
1713, 16sylibrd 262 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
18 ne0i 4249 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
19 on0eln0 6268 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2018, 19syl5ibr 249 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
2120adantr 484 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ 𝐵))
22 oen0 8314 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵o 𝐶))
2322ex 416 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 → ∅ ∈ (𝐵o 𝐶)))
2421, 23syld 47 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ (𝐵o 𝐶)))
25 omordi 8294 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ (𝐵o 𝐶) ∈ On) ∧ ∅ ∈ (𝐵o 𝐶)) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
268, 25syldanl 605 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ (𝐵o 𝐶)) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
2726ex 416 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵o 𝐶) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵))))
2827com23 86 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (∅ ∈ (𝐵o 𝐶) → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵))))
2924, 28mpdd 43 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
30293adant1 1132 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
31 oesuc 8254 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o suc 𝐶) = ((𝐵o 𝐶) ·o 𝐵))
32313adant1 1132 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o suc 𝐶) = ((𝐵o 𝐶) ·o 𝐵))
3332eleq2d 2823 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶) ↔ ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
3430, 33sylibrd 262 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶)))
3517, 34jcad 516 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶))))
36353expa 1120 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶))))
37 sucelon 7596 . . . . . . 7 (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
38 oecl 8264 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴o suc 𝐶) ∈ On)
39 oecl 8264 . . . . . . . . 9 ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵o suc 𝐶) ∈ On)
40 ontr2 6260 . . . . . . . . 9 (((𝐴o suc 𝐶) ∈ On ∧ (𝐵o suc 𝐶) ∈ On) → (((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶)) → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))
4138, 39, 40syl2an 599 . . . . . . . 8 (((𝐴 ∈ On ∧ suc 𝐶 ∈ On) ∧ (𝐵 ∈ On ∧ suc 𝐶 ∈ On)) → (((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶)) → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))
4241anandirs 679 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ suc 𝐶 ∈ On) → (((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶)) → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))
4337, 42sylan2b 597 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶)) → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))
4436, 43syld 47 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))
4544exp31 423 . . . 4 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴𝐵 → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))))
4645com4l 92 . . 3 (𝐵 ∈ On → (𝐶 ∈ On → (𝐴𝐵 → (𝐴 ∈ On → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))))
4746imp 410 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 ∈ On → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶))))
483, 47mpdd 43 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o suc 𝐶) ∈ (𝐵o suc 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wss 3866  c0 4237  Oncon0 6213  suc csuc 6215  (class class class)co 7213   ·o comu 8200  o coe 8201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-oexp 8208
This theorem is referenced by: (None)
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