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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 4592 | . . 3 ⊢ ω ∈ V | |
2 | eqeq1 2184 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅)) | |
3 | fveq2 5515 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (inr‘𝑦) = (inr‘𝑥)) | |
4 | unieq 3818 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥) | |
5 | 4 | fveq2d 5519 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (inl‘∪ 𝑦) = (inl‘∪ 𝑥)) |
6 | 2, 3, 5 | ifbieq12d 3560 | . . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦)) = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))) |
7 | 6 | cbvmptv 4099 | . . . 4 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))) |
8 | suceq 4402 | . . . . 5 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥) | |
9 | 8 | cbvmptv 4099 | . . . 4 ⊢ (𝑦 ∈ ω ↦ suc 𝑦) = (𝑥 ∈ ω ↦ suc 𝑥) |
10 | eqid 2177 | . . . 4 ⊢ case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o)) = case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o)) | |
11 | 7, 9, 10 | omp1eomlem 7092 | . . 3 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))):ω–1-1-onto→(ω ⊔ 1o) |
12 | f1oeng 6756 | . . 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 426 | . 2 ⊢ ω ≈ (ω ⊔ 1o) |
14 | 13 | ensymi 6781 | 1 ⊢ (ω ⊔ 1o) ≈ ω |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∅c0 3422 ifcif 3534 ∪ cuni 3809 class class class wbr 4003 ↦ cmpt 4064 I cid 4288 suc csuc 4365 ωcom 4589 ↾ cres 4628 –1-1-onto→wf1o 5215 ‘cfv 5216 1oc1o 6409 ≈ cen 6737 ⊔ cdju 7035 inlcinl 7043 inrcinr 7044 casecdjucase 7081 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-1st 6140 df-2nd 6141 df-1o 6416 df-er 6534 df-en 6740 df-dju 7036 df-inl 7045 df-inr 7046 df-case 7082 |
This theorem is referenced by: difinfsn 7098 sbthom 14710 |
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