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Theorem omp1eom 7094
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 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)
54fveq2d 5520 . . . . . 6 (𝑦 = 𝑥 → (inl‘ 𝑦) = (inl‘ 𝑥))
62, 3, 5ifbieq12d 3561 . . . . 5 (𝑦 = 𝑥 → if(𝑦 = ∅, (inr‘𝑦), (inl‘ 𝑦)) = if(𝑥 = ∅, (inr‘𝑥), (inl‘ 𝑥)))
76cbvmptv 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)
98cbvmptv 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))
117, 9, 10omp1eomlem 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))
131, 11, 12mp2an 426 . 2 ω ≈ (ω ⊔ 1o)
1413ensymi 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-ontowf1o 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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