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Theorem oeordsuc 8206
 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 6185 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 416 . . 3 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
32adantr 484 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵𝐴 ∈ On))
4 oewordri 8204 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
543adant1 1127 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴o 𝐶) ⊆ (𝐵o 𝐶)))
6 oecl 8148 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
763adant2 1128 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o 𝐶) ∈ On)
8 oecl 8148 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o 𝐶) ∈ On)
983adant1 1127 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o 𝐶) ∈ On)
10 simp1 1133 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 omwordri 8184 . . . . . . . . . . 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 1368 . . . . . . . . . 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 8138 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
15143adant2 1128 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴o suc 𝐶) = ((𝐴o 𝐶) ·o 𝐴))
1615sseq1d 3946 . . . . . . . . 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 4250 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
19 on0eln0 6215 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2018, 19syl5ibr 249 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
2120adantr 484 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ 𝐵))
22 oen0 8198 . . . . . . . . . . . . 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 8178 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ (𝐵o 𝐶) ∈ On) ∧ ∅ ∈ (𝐵o 𝐶)) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
268, 25syldanl 604 . . . . . . . . . . . . 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 1127 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵o 𝐶) ·o 𝐴) ∈ ((𝐵o 𝐶) ·o 𝐵)))
31 oesuc 8138 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o suc 𝐶) = ((𝐵o 𝐶) ·o 𝐵))
32313adant1 1127 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵o suc 𝐶) = ((𝐵o 𝐶) ·o 𝐵))
3332eleq2d 2875 . . . . . . . . 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 1115 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴o suc 𝐶) ⊆ ((𝐵o 𝐶) ·o 𝐴) ∧ ((𝐵o 𝐶) ·o 𝐴) ∈ (𝐵o suc 𝐶))))
37 sucelon 7515 . . . . . . 7 (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
38 oecl 8148 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴o suc 𝐶) ∈ On)
39 oecl 8148 . . . . . . . . 9 ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵o suc 𝐶) ∈ On)
40 ontr2 6207 . . . . . . . . 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 598 . . . . . . . 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 678 . . . . . . 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 596 . . . . . 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 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ⊆ wss 3881  ∅c0 4243  Oncon0 6160  suc csuc 6162  (class class class)co 7136   ·o comu 8086   ↑o coe 8087 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-omul 8093  df-oexp 8094 This theorem is referenced by: (None)
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