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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 6463 | . . . . 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 1004 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐶 ∈ On) | |
4 | simpl1 1002 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ On) | |
5 | simpl2 1003 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On) | |
6 | simpr 110 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
7 | 2, 3, 4, 5, 6 | rdgss 6402 | . . 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 306 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∈ On ∧ 𝐴 ∈ On)) |
9 | oav 6473 | . . . 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 306 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∈ On ∧ 𝐵 ∈ On)) |
12 | oav 6473 | . . . 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 3215 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)) |
15 | 14 | ex 115 | 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 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 ↦ cmpt 4079 Oncon0 4378 suc csuc 4380 Fn wfn 5226 ‘cfv 5231 (class class class)co 5891 reccrdg 6388 +o coa 6432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-suc 4386 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-recs 6324 df-irdg 6389 df-oadd 6439 |
This theorem is referenced by: oaword1 6490 |
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