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Theorem oawordi 6160
Description: Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)
Assertion
Ref Expression
oawordi ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Proof of Theorem oawordi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oafnex 6135 . . . . 5 (𝑥 ∈ V ↦ suc 𝑥) Fn V
21a1i 9 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝑥 ∈ V ↦ suc 𝑥) Fn V)
3 simpl3 944 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → 𝐶 ∈ On)
4 simpl1 942 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → 𝐴 ∈ On)
5 simpl2 943 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → 𝐵 ∈ On)
6 simpr 108 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → 𝐴𝐵)
72, 3, 4, 5, 6rdgss 6078 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐴) ⊆ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐵))
83, 4jca 300 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐶 ∈ On ∧ 𝐴 ∈ On))
9 oav 6145 . . . 4 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +𝑜 𝐴) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐴))
108, 9syl 14 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐴))
113, 5jca 300 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐶 ∈ On ∧ 𝐵 ∈ On))
12 oav 6145 . . . 4 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐵))
1311, 12syl 14 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐵))
147, 10, 133sstr4d 3053 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))
1514ex 113 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  Vcvv 2612  wss 2984  cmpt 3865  Oncon0 4153  suc csuc 4155   Fn wfn 4962  cfv 4967  (class class class)co 5589  reccrdg 6064   +𝑜 coa 6108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-recs 6000  df-irdg 6065  df-oadd 6115
This theorem is referenced by:  oaword1  6162
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