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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 4593 | . . 3 ⊢ ω ∈ V | |
2 | eqeq1 2184 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅)) | |
3 | fveq2 5516 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (inr‘𝑦) = (inr‘𝑥)) | |
4 | unieq 3819 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥) | |
5 | 4 | fveq2d 5520 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (inl‘∪ 𝑦) = (inl‘∪ 𝑥)) |
6 | 2, 3, 5 | ifbieq12d 3561 | . . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦)) = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))) |
7 | 6 | cbvmptv 4100 | . . . 4 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥))) |
8 | suceq 4403 | . . . . 5 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥) | |
9 | 8 | cbvmptv 4100 | . . . 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 7093 | . . 3 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))):ω–1-1-onto→(ω ⊔ 1o) |
12 | f1oeng 6757 | . . 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 6782 | 1 ⊢ (ω ⊔ 1o) ≈ ω |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 Vcvv 2738 ∅c0 3423 ifcif 3535 ∪ cuni 3810 class class class wbr 4004 ↦ cmpt 4065 I cid 4289 suc csuc 4366 ωcom 4590 ↾ cres 4629 –1-1-onto→wf1o 5216 ‘cfv 5217 1oc1o 6410 ≈ cen 6738 ⊔ cdju 7036 inlcinl 7044 inrcinr 7045 casecdjucase 7082 |
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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 |
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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-iord 4367 df-on 4369 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-1st 6141 df-2nd 6142 df-1o 6417 df-er 6535 df-en 6741 df-dju 7037 df-inl 7046 df-inr 7047 df-case 7083 |
This theorem is referenced by: difinfsn 7099 sbthom 14777 |
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