Theorem omp1eom 7429

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.

Step	Hyp	Ref	Expression
1		omex 4738	. . 3 ⊢ ω ∈ V
2		eqeq1 2245	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
3		fveq2 5693	. . . . . 6 ⊢ (𝑦 = 𝑥 → (inr‘𝑦) = (inr‘𝑥))
4		unieq 3942	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
5	4	fveq2d 5697	. . . . . 6 ⊢ (𝑦 = 𝑥 → (inl‘∪ 𝑦) = (inl‘∪ 𝑥))
6	2, 3, 5	ifbieq12d 3667	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦)) = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
7	6	cbvmptv 4225	. . . 4 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
8		suceq 4545	. . . . 5 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
9	8	cbvmptv 4225	. . . 4 ⊢ (𝑦 ∈ ω ↦ suc 𝑦) = (𝑥 ∈ ω ↦ suc 𝑥)
10		eqid 2238	. . . 4 ⊢ case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o)) = case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o))
11	7, 9, 10	omp1eomlem 7428	. . 3 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))):ω–1-1-onto→(ω ⊔ 1o)
12		f1oeng 7037	. . 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 430	. 2 ⊢ ω ≈ (ω ⊔ 1o)
14	13	ensymi 7063	1 ⊢ (ω ⊔ 1o) ≈ ω

Syntax hints:   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  ifcif 3638  ∪ cuni 3933   class class class wbr 4128   ↦ cmpt 4190   I cid 4431  suc csuc 4508  ωcom 4735   ↾ cres 4774  –1-1-onto→wf1o 5374  ‘cfv 5375  1_oc1o 6674   ≈ cen 7014   ⊔ cdju 7371  inlcinl 7379  inrcinr 7380  casecdjucase 7417

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1st 6368  df-2nd 6369  df-1o 6681  df-er 6801  df-en 7017  df-dju 7372  df-inl 7381  df-inr 7382  df-case 7418

This theorem is referenced by:  difinfsn  7434  sbthom  17045