Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  omp1eom GIF version

Theorem omp1eom 6930
 Description: Adding one to ω. (Contributed by Jim Kingdon, 10-Jul-2023.)
Assertion
Ref Expression
omp1eom (ω ⊔ 1o) ≈ ω

Proof of Theorem omp1eom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 4465 . . 3 ω ∈ V
2 eqeq1 2119 . . . . . 6 (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
3 fveq2 5373 . . . . . 6 (𝑦 = 𝑥 → (inr‘𝑦) = (inr‘𝑥))
4 unieq 3709 . . . . . . 7 (𝑦 = 𝑥 𝑦 = 𝑥)
54fveq2d 5377 . . . . . 6 (𝑦 = 𝑥 → (inl‘ 𝑦) = (inl‘ 𝑥))
62, 3, 5ifbieq12d 3462 . . . . 5 (𝑦 = 𝑥 → if(𝑦 = ∅, (inr‘𝑦), (inl‘ 𝑦)) = if(𝑥 = ∅, (inr‘𝑥), (inl‘ 𝑥)))
76cbvmptv 3982 . . . 4 (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘ 𝑦))) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘ 𝑥)))
8 suceq 4282 . . . . 5 (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
98cbvmptv 3982 . . . 4 (𝑦 ∈ ω ↦ suc 𝑦) = (𝑥 ∈ ω ↦ suc 𝑥)
10 eqid 2113 . . . 4 case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o)) = case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o))
117, 9, 10omp1eomlem 6929 . . 3 (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘ 𝑦))):ω–1-1-onto→(ω ⊔ 1o)
12 f1oeng 6603 . . 3 ((ω ∈ V ∧ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘ 𝑦))):ω–1-1-onto→(ω ⊔ 1o)) → ω ≈ (ω ⊔ 1o))
131, 11, 12mp2an 420 . 2 ω ≈ (ω ⊔ 1o)
1413ensymi 6628 1 (ω ⊔ 1o) ≈ ω
 Colors of variables: wff set class Syntax hints:   = wceq 1312   ∈ wcel 1461  Vcvv 2655  ∅c0 3327  ifcif 3438  ∪ cuni 3700   class class class wbr 3893   ↦ cmpt 3947   I cid 4168  suc csuc 4245  ωcom 4462   ↾ cres 4499  –1-1-onto→wf1o 5078  ‘cfv 5079  1oc1o 6258   ≈ cen 6584   ⊔ cdju 6872  inlcinl 6880  inrcinr 6881  casecdjucase 6918 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460 This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-1st 5990  df-2nd 5991  df-1o 6265  df-er 6381  df-en 6587  df-dju 6873  df-inl 6882  df-inr 6883  df-case 6919 This theorem is referenced by:  difinfsn  6935  sbthom  12902
 Copyright terms: Public domain W3C validator