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Theorem oawordri 7590
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 6623 . . . . . 6 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
2 oveq2 6623 . . . . . 6 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
31, 2sseq12d 3619 . . . . 5 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅)))
4 oveq2 6623 . . . . . 6 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
5 oveq2 6623 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
64, 5sseq12d 3619 . . . . 5 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦)))
7 oveq2 6623 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
8 oveq2 6623 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
97, 8sseq12d 3619 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
10 oveq2 6623 . . . . . 6 (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
11 oveq2 6623 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1210, 11sseq12d 3619 . . . . 5 (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
13 oa0 7556 . . . . . . . 8 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
1413adantr 481 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 ∅) = 𝐴)
15 oa0 7556 . . . . . . . 8 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
1615adantl 482 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 ∅) = 𝐵)
1714, 16sseq12d 3619 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅) ↔ 𝐴𝐵))
1817biimpar 502 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) ⊆ (𝐵 +𝑜 ∅))
19 oacl 7575 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 𝑦) ∈ On)
20 eloni 5702 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝑦) ∈ On → Ord (𝐴 +𝑜 𝑦))
2119, 20syl 17 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → Ord (𝐴 +𝑜 𝑦))
22 oacl 7575 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
23 eloni 5702 . . . . . . . . . . . 12 ((𝐵 +𝑜 𝑦) ∈ On → Ord (𝐵 +𝑜 𝑦))
2422, 23syl 17 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → Ord (𝐵 +𝑜 𝑦))
25 ordsucsssuc 6985 . . . . . . . . . . 11 ((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 494 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
2726anandirs 873 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ⊆ suc (𝐵 +𝑜 𝑦)))
28 oasuc 7564 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
2928adantlr 750 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
30 oasuc 7564 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3130adantll 749 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3229, 31sseq12d 3619 . . . . . . . . 9 (((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 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
3433biimpd 219 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦)))
3534expcom 451 . . . . . 6 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))))
3635adantrd 484 . . . . 5 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ⊆ (𝐵 +𝑜 suc 𝑦))))
37 vex 3193 . . . . . . . 8 𝑥 ∈ V
38 ss2iun 4509 . . . . . . . . 9 (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → 𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵 +𝑜 𝑦))
39 oalim 7572 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +𝑜 𝑥) = 𝑦𝑥 (𝐴 +𝑜 𝑦))
4039adantlr 750 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +𝑜 𝑥) = 𝑦𝑥 (𝐴 +𝑜 𝑦))
41 oalim 7572 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
4241adantll 749 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +𝑜 𝑥) = 𝑦𝑥 (𝐵 +𝑜 𝑦))
4340, 42sseq12d 3619 . . . . . . . . 9 (((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 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)))
4537, 44mpanr1 718 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥)))
4645expcom 451 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 (𝐴 +𝑜 𝑦) ⊆ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑥) ⊆ (𝐵 +𝑜 𝑥))))
4746adantrd 484 . . . . 5 (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 7026 . . . 4 (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))
4948exp4c 635 . . 3 (𝐶 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))))
5049com3l 89 . 2 (𝐴 ∈ On → (𝐵 ∈ On → (𝐶 ∈ On → (𝐴𝐵 → (𝐴 +𝑜 𝐶) ⊆ (𝐵 +𝑜 𝐶)))))
51503imp 1254 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  wss 3560  c0 3897   ciun 4492  Ord word 5691  Oncon0 5692  Lim wlim 5693  suc csuc 5694  (class class class)co 6615   +𝑜 coa 7517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524
This theorem is referenced by:  oaword2  7593  omwordri  7612  oaabs2  7685
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