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Mirrors > Home > ILE Home > Th. List > oawordi | GIF version |
Description: Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.) |
Ref | Expression |
---|---|
oawordi | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oafnex 6308 | . . . . 5 ⊢ (𝑥 ∈ V ↦ suc 𝑥) Fn V | |
2 | 1 | a1i 9 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ V ↦ suc 𝑥) Fn V) |
3 | simpl3 971 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐶 ∈ On) | |
4 | simpl1 969 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ On) | |
5 | simpl2 970 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On) | |
6 | simpr 109 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
7 | 2, 3, 4, 5, 6 | rdgss 6248 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐴) ⊆ (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐵)) |
8 | 3, 4 | jca 304 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∈ On ∧ 𝐴 ∈ On)) |
9 | oav 6318 | . . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐴)) | |
10 | 8, 9 | syl 14 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 +o 𝐴) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐴)) |
11 | 3, 5 | jca 304 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∈ On ∧ 𝐵 ∈ On)) |
12 | oav 6318 | . . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐵)) | |
13 | 11, 12 | syl 14 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 +o 𝐵) = (rec((𝑥 ∈ V ↦ suc 𝑥), 𝐶)‘𝐵)) |
14 | 7, 10, 13 | 3sstr4d 3112 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)) |
15 | 14 | ex 114 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 947 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ⊆ wss 3041 ↦ cmpt 3959 Oncon0 4255 suc csuc 4257 Fn wfn 5088 ‘cfv 5093 (class class class)co 5742 reccrdg 6234 +o coa 6278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-recs 6170 df-irdg 6235 df-oadd 6285 |
This theorem is referenced by: oaword1 6335 |
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