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Theorem oeordsuc 8632
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 6409 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 412 . . 3 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
32adantr 480 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵𝐴 ∈ On))
4 oewordri 8630 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
543adant1 1131 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
6 oecl 8575 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
763adant2 1132 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
8 oecl 8575 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o 𝐶) ∈ On)
983adant1 1131 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o 𝐶) ∈ On)
10 simp1 1137 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 omwordri 8610 . . . . . . . . . . 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 8565 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
15143adant2 1132 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
1615sseq1d 4015 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ↔ ((𝐴o 𝐶) ·o 𝐴) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
1713, 16sylibrd 259 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
18 ne0i 4341 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
19 on0eln0 6440 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2018, 19imbitrrid 246 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
2120adantr 480 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ 𝐵))
22 oen0 8624 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵o 𝐶))
2322ex 412 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 → ∅ ∈ (𝐵o 𝐶)))
2421, 23syld 47 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ (𝐵o 𝐶)))
25 omordi 8604 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ (𝐵o 𝐶) ∈ On) ∧ ∅ ∈ (𝐵o 𝐶)) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
268, 25syldanl 602 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ (𝐵o 𝐶)) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
2726ex 412 . . . . . . . . . . . 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 1131 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
31 oesuc 8565 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o suc 𝐶) = ((𝐵o 𝐶) ·o 𝐵))
32313adant1 1131 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o suc 𝐶) = ((𝐵o 𝐶) ·o 𝐵))
3332eleq2d 2827 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶) ↔ ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
3430, 33sylibrd 259 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶)))
3517, 34jcad 512 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶))))
36353expa 1119 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶))))
37 onsucb 7837 . . . . . . 7 (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
38 oecl 8575 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴o suc 𝐶) ∈ On)
39 oecl 8575 . . . . . . . . 9 ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵o suc 𝐶) ∈ On)
40 ontr2 6431 . . . . . . . . 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 596 . . . . . . . 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 594 . . . . . 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 419 . . . 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 406 . 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 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wss 3951  c0 4333  Oncon0 6384  suc csuc 6386  (class class class)co 7431   ·o comu 8504  o coe 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-oexp 8512
This theorem is referenced by: (None)
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