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Mirrors > Home > ILE Home > Th. List > omp1eom | GIF version |
Description: Adding one to ω. (Contributed by Jim Kingdon, 10-Jul-2023.) |
Ref | Expression |
---|---|
omp1eom | ⊢ (ω ⊔ 1o) ≈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4507 | . . 3 ⊢ ω ∈ V | |
2 | eqeq1 2146 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅)) | |
3 | fveq2 5421 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (inr‘𝑦) = (inr‘𝑥)) | |
4 | unieq 3745 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥) | |
5 | 4 | fveq2d 5425 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (inl‘∪ 𝑦) = (inl‘∪ 𝑥)) |
6 | 2, 3, 5 | ifbieq12d 3498 | . . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦)) = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))) |
7 | 6 | cbvmptv 4024 | . . . 4 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))) |
8 | suceq 4324 | . . . . 5 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥) | |
9 | 8 | cbvmptv 4024 | . . . 4 ⊢ (𝑦 ∈ ω ↦ suc 𝑦) = (𝑥 ∈ ω ↦ suc 𝑥) |
10 | eqid 2139 | . . . 4 ⊢ case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o)) = case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o)) | |
11 | 7, 9, 10 | omp1eomlem 6979 | . . 3 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))):ω–1-1-onto→(ω ⊔ 1o) |
12 | f1oeng 6651 | . . 3 ⊢ ((ω ∈ V ∧ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))):ω–1-1-onto→(ω ⊔ 1o)) → ω ≈ (ω ⊔ 1o)) | |
13 | 1, 11, 12 | mp2an 422 | . 2 ⊢ ω ≈ (ω ⊔ 1o) |
14 | 13 | ensymi 6676 | 1 ⊢ (ω ⊔ 1o) ≈ ω |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∅c0 3363 ifcif 3474 ∪ cuni 3736 class class class wbr 3929 ↦ cmpt 3989 I cid 4210 suc csuc 4287 ωcom 4504 ↾ cres 4541 –1-1-onto→wf1o 5122 ‘cfv 5123 1oc1o 6306 ≈ cen 6632 ⊔ cdju 6922 inlcinl 6930 inrcinr 6931 casecdjucase 6968 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-er 6429 df-en 6635 df-dju 6923 df-inl 6932 df-inr 6933 df-case 6969 |
This theorem is referenced by: difinfsn 6985 sbthom 13221 |
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