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Theorem oeordsuc 7719
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 5786 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 449 . . 3 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
32adantr 480 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵𝐴 ∈ On))
4 oewordri 7717 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
543adant1 1099 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
6 oecl 7662 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
763adant2 1100 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
8 oecl 7662 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 𝐶) ∈ On)
983adant1 1099 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 𝐶) ∈ On)
10 simp1 1081 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 omwordri 7697 . . . . . . . . . . 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 1366 . . . . . . . . . 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 7652 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
15143adant2 1100 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
1615sseq1d 3665 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ↔ ((𝐴𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
1713, 16sylibrd 249 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
18 ne0i 3954 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
19 on0eln0 5818 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2018, 19syl5ibr 236 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
2120adantr 480 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ 𝐵))
22 oen0 7711 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵𝑜 𝐶))
2322ex 449 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 → ∅ ∈ (𝐵𝑜 𝐶)))
2421, 23syld 47 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ (𝐵𝑜 𝐶)))
25 simpl 472 . . . . . . . . . . . . . . 15 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ∈ On)
2625, 8jca 553 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ On ∧ (𝐵𝑜 𝐶) ∈ On))
27 omordi 7691 . . . . . . . . . . . . . 14 (((𝐵 ∈ On ∧ (𝐵𝑜 𝐶) ∈ On) ∧ ∅ ∈ (𝐵𝑜 𝐶)) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
2826, 27sylan 487 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ (𝐵𝑜 𝐶)) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
2928ex 449 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ (𝐵𝑜 𝐶) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵))))
3029com23 86 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (∅ ∈ (𝐵𝑜 𝐶) → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵))))
3124, 30mpdd 43 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
32313adant1 1099 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
33 oesuc 7652 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) = ((𝐵𝑜 𝐶) ·𝑜 𝐵))
34333adant1 1099 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) = ((𝐵𝑜 𝐶) ·𝑜 𝐵))
3534eleq2d 2716 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶) ↔ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
3632, 35sylibrd 249 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶)))
3717, 36jcad 554 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶))))
38373expa 1284 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶))))
39 sucelon 7059 . . . . . . 7 (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
40 oecl 7662 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) ∈ On)
41 oecl 7662 . . . . . . . . 9 ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) ∈ On)
42 ontr2 5810 . . . . . . . . 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)))
4340, 41, 42syl2an 493 . . . . . . . 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)))
4443anandirs 891 . . . . . . 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)))
4539, 44sylan2b 491 . . . . . 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)))
4638, 45syld 47 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
4746exp31 629 . . . 4 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))))
4847com4l 92 . . 3 (𝐵 ∈ On → (𝐶 ∈ On → (𝐴𝐵 → (𝐴 ∈ On → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))))
4948imp 444 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 ∈ On → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶))))
503, 49mpdd 43 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wss 3607  c0 3948  Oncon0 5761  suc csuc 5763  (class class class)co 6690   ·𝑜 comu 7603  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-oexp 7611
This theorem is referenced by: (None)
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