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Theorem oeordsuc 8209
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 6209 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 413 . . 3 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
32adantr 481 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵𝐴 ∈ On))
4 oewordri 8207 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
543adant1 1122 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
6 oecl 8151 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
763adant2 1123 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
8 oecl 8151 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o 𝐶) ∈ On)
983adant1 1122 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o 𝐶) ∈ On)
10 simp1 1128 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 omwordri 8187 . . . . . . . . . . 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 1363 . . . . . . . . . 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 8141 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
15143adant2 1123 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
1615sseq1d 3995 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ↔ ((𝐴o 𝐶) ·o 𝐴) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
1713, 16sylibrd 260 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴)))
18 ne0i 4297 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
19 on0eln0 6239 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2018, 19syl5ibr 247 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
2120adantr 481 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ 𝐵))
22 oen0 8201 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵o 𝐶))
2322ex 413 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 → ∅ ∈ (𝐵o 𝐶)))
2421, 23syld 47 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ (𝐵o 𝐶)))
25 omordi 8181 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ (𝐵o 𝐶) ∈ On) ∧ ∅ ∈ (𝐵o 𝐶)) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
268, 25syldanl 601 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ (𝐵o 𝐶)) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
2726ex 413 . . . . . . . . . . . 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 1122 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
31 oesuc 8141 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o suc 𝐶) = ((𝐵o 𝐶) ·o 𝐵))
32313adant1 1122 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o suc 𝐶) = ((𝐵o 𝐶) ·o 𝐵))
3332eleq2d 2895 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶) ↔ ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
3430, 33sylibrd 260 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶)))
3517, 34jcad 513 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶))))
36353expa 1110 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶))))
37 sucelon 7521 . . . . . . 7 (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
38 oecl 8151 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴o suc 𝐶) ∈ On)
39 oecl 8151 . . . . . . . . 9 ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵o suc 𝐶) ∈ On)
40 ontr2 6231 . . . . . . . . 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 595 . . . . . . . 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 675 . . . . . . 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 593 . . . . . 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 420 . . . 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 407 . 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 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wss 3933  c0 4288  Oncon0 6184  suc csuc 6186  (class class class)co 7145   ·o comu 8089  o coe 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-oexp 8097
This theorem is referenced by: (None)
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