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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 6135 | . . . . 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 944 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐶 ∈ On) | |
4 | simpl1 942 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ On) | |
5 | simpl2 943 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On) | |
6 | simpr 108 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
7 | 2, 3, 4, 5, 6 | rdgss 6078 | . . 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 300 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∈ On ∧ 𝐴 ∈ On)) |
9 | oav 6145 | . . . 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 300 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ∈ On ∧ 𝐵 ∈ On)) |
12 | oav 6145 | . . . 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 3053 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)) |
15 | 14 | ex 113 | 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 102 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 Vcvv 2612 ⊆ wss 2984 ↦ cmpt 3865 Oncon0 4153 suc csuc 4155 Fn wfn 4962 ‘cfv 4967 (class class class)co 5589 reccrdg 6064 +𝑜 coa 6108 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-iord 4156 df-on 4158 df-suc 4161 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-recs 6000 df-irdg 6065 df-oadd 6115 |
This theorem is referenced by: oaword1 6162 |
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