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Mirrors > Home > ILE Home > Th. List > oawordi | Structured version 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 5934 | . . . . 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 897 | . . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → 𝐶 ∈ On) | |
4 | simpl1 895 | . . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → A ∈ On) | |
5 | simpl2 896 | . . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → B ∈ On) | |
6 | ax-ia2 100 | . . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → A ⊆ B) | |
7 | 2, 3, 4, 5, 6 | rdgss 5885 | . . 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 290 | . . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (𝐶 ∈ On ∧ A ∈ On)) |
9 | oav 5944 | . . . 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 290 | . . . 4 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (𝐶 ∈ On ∧ B ∈ On)) |
12 | oav 5944 | . . . 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 2964 | . 2 ⊢ (((A ∈ On ∧ B ∈ On ∧ 𝐶 ∈ On) ∧ A ⊆ B) → (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)) |
15 | 14 | ex 108 | 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 97 ∧ w3a 873 = wceq 1228 ∈ wcel 1375 Vcvv 2534 ⊆ wss 2893 ↦ cmpt 3791 Oncon0 4047 suc csuc 4049 Fn wfn 4822 ‘cfv 4827 (class class class)co 5434 reccrdg 5872 +𝑜 coa 5908 |
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 1316 ax-7 1317 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 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 2005 ax-coll 3845 ax-sep 3848 ax-pow 3900 ax-pr 3917 ax-un 4118 ax-setind 4202 |
This theorem depends on definitions: df-bi 110 df-3an 875 df-tru 1231 df-fal 1234 df-nf 1330 df-sb 1629 df-eu 1886 df-mo 1887 df-clab 2010 df-cleq 2016 df-clel 2019 df-nfc 2150 df-ne 2189 df-ral 2288 df-rex 2289 df-reu 2290 df-rab 2292 df-v 2536 df-sbc 2741 df-csb 2829 df-dif 2896 df-un 2898 df-in 2900 df-ss 2907 df-nul 3201 df-pw 3335 df-sn 3355 df-pr 3356 df-op 3358 df-uni 3554 df-iun 3632 df-br 3738 df-opab 3792 df-mpt 3793 df-tr 3828 df-id 4003 df-iord 4050 df-on 4052 df-suc 4055 df-xp 4276 df-rel 4277 df-cnv 4278 df-co 4279 df-dm 4280 df-rn 4281 df-res 4282 df-ima 4283 df-iota 4792 df-fun 4829 df-fn 4830 df-f 4831 df-f1 4832 df-fo 4833 df-f1o 4834 df-fv 4835 df-ov 5437 df-oprab 5438 df-mpt2 5439 df-recs 5837 df-irdg 5873 df-oadd 5915 |
This theorem is referenced by: oaword1 5960 |
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