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Theorem oawordri 8159
 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 7143 . . . . 5 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2 oveq2 7143 . . . . 5 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
31, 2sseq12d 3948 . . . 4 (𝑥 = ∅ → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ⊆ (𝐵 +o ∅)))
4 oveq2 7143 . . . . 5 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
5 oveq2 7143 . . . . 5 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
64, 5sseq12d 3948 . . . 4 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦)))
7 oveq2 7143 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
8 oveq2 7143 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
97, 8sseq12d 3948 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
10 oveq2 7143 . . . . 5 (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
11 oveq2 7143 . . . . 5 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1210, 11sseq12d 3948 . . . 4 (𝑥 = 𝐶 → ((𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
13 oa0 8124 . . . . . . 7 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
1413adantr 484 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o ∅) = 𝐴)
15 oa0 8124 . . . . . . 7 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
1615adantl 485 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o ∅) = 𝐵)
1714, 16sseq12d 3948 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o ∅) ⊆ (𝐵 +o ∅) ↔ 𝐴𝐵))
1817biimpar 481 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 +o ∅) ⊆ (𝐵 +o ∅))
19 oacl 8143 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o 𝑦) ∈ On)
20 eloni 6169 . . . . . . . . . . 11 ((𝐴 +o 𝑦) ∈ On → Ord (𝐴 +o 𝑦))
2119, 20syl 17 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → Ord (𝐴 +o 𝑦))
22 oacl 8143 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
23 eloni 6169 . . . . . . . . . . 11 ((𝐵 +o 𝑦) ∈ On → Ord (𝐵 +o 𝑦))
2422, 23syl 17 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → Ord (𝐵 +o 𝑦))
25 ordsucsssuc 7518 . . . . . . . . . 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 598 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
2726anandirs 678 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ suc (𝐴 +o 𝑦) ⊆ suc (𝐵 +o 𝑦)))
28 oasuc 8132 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
2928adantlr 714 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
30 oasuc 8132 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3130adantll 713 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3229, 31sseq12d 3948 . . . . . . . 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 285 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) ↔ (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
3433biimpd 232 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦)))
3534expcom 417 . . . . 5 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))))
3635adantrd 495 . . . 4 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ⊆ (𝐵 +o suc 𝑦))))
37 vex 3444 . . . . . . 7 𝑥 ∈ V
38 ss2iun 4899 . . . . . . . 8 (∀𝑦𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → 𝑦𝑥 (𝐴 +o 𝑦) ⊆ 𝑦𝑥 (𝐵 +o 𝑦))
39 oalim 8140 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = 𝑦𝑥 (𝐴 +o 𝑦))
4039adantlr 714 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 +o 𝑥) = 𝑦𝑥 (𝐴 +o 𝑦))
41 oalim 8140 . . . . . . . . . 10 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
4241adantll 713 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 +o 𝑥) = 𝑦𝑥 (𝐵 +o 𝑦))
4340, 42sseq12d 3948 . . . . . . . 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 249 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∀𝑦𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)))
4537, 44mpanr1 702 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥)))
4645expcom 417 . . . . 5 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 (𝐴 +o 𝑦) ⊆ (𝐵 +o 𝑦) → (𝐴 +o 𝑥) ⊆ (𝐵 +o 𝑥))))
4746adantrd 495 . . . 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 7559 . . 3 (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))
4948exp4c 436 . 2 (𝐶 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶)))))
50493imp231 1110 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∪ ciun 4881  Ord word 6158  Oncon0 6159  Lim wlim 6160  suc csuc 6161  (class class class)co 7135   +o coa 8082 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 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 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 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089 This theorem is referenced by:  oaword2  8162  omwordri  8181  oaabs2  8255
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