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Theorem oawordi 5924
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 5904 . . . . 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 899 . . . 4 (((A On B On 𝐶 On) AB) → 𝐶 On)
4 simpl1 897 . . . 4 (((A On B On 𝐶 On) AB) → A On)
5 simpl2 898 . . . 4 (((A On B On 𝐶 On) AB) → B On)
6 ax-ia2 100 . . . 4 (((A On B On 𝐶 On) AB) → AB)
72, 3, 4, 5, 6rdgss 5855 . . 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 5913 . . . 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 5913 . . . 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 2966 . 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 875   = wceq 1373   wcel 1375  Vcvv 2533  wss 2895   cmpt 3770  Oncon0 4024  suc csuc 4026   Fn wfn 4791  cfv 4796  (class class class)co 5405  reccrdg 5842   +𝑜 coa 5878
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 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-coll 3824  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093  ax-setind 4174
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-fal 1232  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rex 2288  df-reu 2289  df-rab 2291  df-v 2535  df-sbc 2740  df-csb 2829  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-tr 3807  df-id 3983  df-iord 4027  df-on 4028  df-suc 4031  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-f 4800  df-f1 4801  df-fo 4802  df-f1o 4803  df-fv 4804  df-ov 5408  df-oprab 5409  df-mpt2 5410  df-recs 5807  df-irdg 5843  df-oadd 5885
This theorem is referenced by: (None)
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