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Theorem oeordsuc 7561
 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 5665 . . . 4 ((𝐵 ∈ On ∧ 𝐴𝐵) → 𝐴 ∈ On)
21ex 449 . . 3 (𝐵 ∈ On → (𝐴𝐵𝐴 ∈ On))
32adantr 480 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵𝐴 ∈ On))
4 oewordri 7559 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
543adant1 1072 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶)))
6 oecl 7504 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
763adant2 1073 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 𝐶) ∈ On)
8 oecl 7504 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 𝐶) ∈ On)
983adant1 1072 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 𝐶) ∈ On)
10 simp1 1054 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
11 omwordri 7539 . . . . . . . . . . 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 1318 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 𝐶) ⊆ (𝐵𝑜 𝐶) → ((𝐴𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
135, 12syld 46 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
14 oesuc 7494 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
15143adant2 1073 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) = ((𝐴𝑜 𝐶) ·𝑜 𝐴))
1615sseq1d 3595 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ↔ ((𝐴𝑜 𝐶) ·𝑜 𝐴) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
1713, 16sylibrd 248 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴)))
18 ne0i 3880 . . . . . . . . . . . . . 14 (𝐴𝐵𝐵 ≠ ∅)
19 on0eln0 5697 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
2018, 19syl5ibr 235 . . . . . . . . . . . . 13 (𝐵 ∈ On → (𝐴𝐵 → ∅ ∈ 𝐵))
2120adantr 480 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ∅ ∈ 𝐵))
22 oen0 7553 . . . . . . . . . . . . 13 (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐵𝑜 𝐶))
2322ex 449 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐵 → ∅ ∈ (𝐵𝑜 𝐶)))
2421, 23syld 46 . . . . . . . . . . 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 7533 . . . . . . . . . . . . . 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 84 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (∅ ∈ (𝐵𝑜 𝐶) → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵))))
3124, 30mpdd 42 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
32313adant1 1072 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
33 oesuc 7494 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) = ((𝐵𝑜 𝐶) ·𝑜 𝐵))
34333adant1 1072 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) = ((𝐵𝑜 𝐶) ·𝑜 𝐵))
3534eleq2d 2673 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶) ↔ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ ((𝐵𝑜 𝐶) ·𝑜 𝐵)))
3632, 35sylibrd 248 . . . . . . . 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 1257 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → ((𝐴𝑜 suc 𝐶) ⊆ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∧ ((𝐵𝑜 𝐶) ·𝑜 𝐴) ∈ (𝐵𝑜 suc 𝐶))))
39 sucelon 6909 . . . . . . 7 (𝐶 ∈ On ↔ suc 𝐶 ∈ On)
40 oecl 7504 . . . . . . . . 9 ((𝐴 ∈ On ∧ suc 𝐶 ∈ On) → (𝐴𝑜 suc 𝐶) ∈ On)
41 oecl 7504 . . . . . . . . 9 ((𝐵 ∈ On ∧ suc 𝐶 ∈ On) → (𝐵𝑜 suc 𝐶) ∈ On)
42 ontr2 5689 . . . . . . . . 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 870 . . . . . . 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 46 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))
4746exp31 628 . . . 4 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴𝐵 → (𝐴𝑜 suc 𝐶) ∈ (𝐵𝑜 suc 𝐶)))))
4847com4l 90 . . 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 42 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 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  Oncon0 5640  suc csuc 5642  (class class class)co 6549   ·𝑜 comu 7445   ↑𝑜 coe 7446 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-oexp 7453 This theorem is referenced by: (None)
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