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Theorem oawordi 6012
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 5987 . . . . 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 909 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → 𝐶 ∈ On)
4 simpl1 907 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → 𝐴 ∈ On)
5 simpl2 908 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → 𝐵 ∈ On)
6 simpr 103 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → 𝐴𝐵)
72, 3, 4, 5, 6rdgss 5933 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐴) ⊆ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐵))
83, 4jca 290 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐶 ∈ On ∧ 𝐴 ∈ On))
9 oav 5997 . . . 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 290 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐶 ∈ On ∧ 𝐵 ∈ On))
12 oav 5997 . . . 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 2985 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))
1514ex 108 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 97  w3a 885   = wceq 1243  wcel 1393  Vcvv 2554  wss 2914  cmpt 3815  Oncon0 4072  suc csuc 4074   Fn wfn 4860  cfv 4865  (class class class)co 5475  reccrdg 5919   +𝑜 coa 5961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-id 4027  df-iord 4075  df-on 4077  df-suc 4080  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-recs 5883  df-irdg 5920  df-oadd 5968
This theorem is referenced by:  oaword1  6013
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