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Theorem oawordri 7675
Description: Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59. (Contributed by NM, 7-Dec-2004.)
Assertion
Ref Expression
oawordri ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))

Proof of Theorem oawordri
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . . . 5 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
2 oveq2 6698 . . . . 5 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
31, 2sseq12d 3667 . . . 4 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
4 oveq2 6698 . . . . 5 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
5 oveq2 6698 . . . . 5 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
64, 5sseq12d 3667 . . . 4 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)))
7 oveq2 6698 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
8 oveq2 6698 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
97, 8sseq12d 3667 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
10 oveq2 6698 . . . . 5 (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
11 oveq2 6698 . . . . 5 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1210, 11sseq12d 3667 . . . 4 (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
13 oa0 7641 . . . . . . 7 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
1413adantr 480 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 ∅) = 𝐴)
15 oa0 7641 . . . . . . 7 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
1615adantl 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 ∅) = 𝐵)
1714, 16sseq12d 3667 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅) ↔ 𝐴𝐵))
1817biimpar 501 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅))
19 oacl 7660 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 𝑦) ∈ On)
20 eloni 5771 . . . . . . . . . . 11 ((𝐴 +𝑜 𝑦) ∈ On → Ord (𝐴 +𝑜 𝑦))
2119, 20syl 17 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → Ord (𝐴 +𝑜 𝑦))
22 oacl 7660 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
23 eloni 5771 . . . . . . . . . . 11 ((𝐵 +𝑜 𝑦) ∈ On → Ord (𝐵 +𝑜 𝑦))
2422, 23syl 17 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → Ord (𝐵 +𝑜 𝑦))
25 ordsucsssuc 7065 . . . . . . . . . 10 ((Ord (𝐴 +𝑜 𝑦) ∧ Ord (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
2621, 24, 25syl2an 493 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
2726anandirs 891 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
28 oasuc 7649 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
2928adantlr 751 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
30 oasuc 7649 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3130adantll 750 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3229, 31sseq12d 3667 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
3327, 32bitr4d 271 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
3433biimpd 219 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
3534expcom 450 . . . . 5 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))))
3635adantrd 483 . . . 4 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))))
37 vex 3234 . . . . . . 7 𝑥 ∈ V
38 ss2iun 4568 . . . . . . . 8 (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → 𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵 +𝑜 𝑦))
39 oalim 7657 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +𝑜 𝑥) = 𝑦𝑥 (𝐴 +𝑜 𝑦))
4039adantlr 751 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +𝑜 𝑥) = 𝑦𝑥 (𝐴 +𝑜 𝑦))
41 oalim 7657 . . . . . . . . . 10 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
4241adantll 750 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
4340, 42sseq12d 3667 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ 𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵 +𝑜 𝑦)))
4438, 43syl5ibr 236 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)))
4537, 44mpanr1 719 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)))
4645expcom 450 . . . . 5 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))))
4746adantrd 483 . . . 4 (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))))
483, 6, 9, 12, 18, 36, 47tfinds3 7106 . . 3 (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
4948exp4c 635 . 2 (𝐶 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))))
50493imp231 1277 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948   ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690   +𝑜 coa 7602
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-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609
This theorem is referenced by:  oaword2  7678  omwordri  7697  oaabs2  7770
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