Theorem omp1eom 7399

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 4720	. . 3 ⊢ ω ∈ V
2		eqeq1 2241	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
3		fveq2 5675	. . . . . 6 ⊢ (𝑦 = 𝑥 → (inr‘𝑦) = (inr‘𝑥))
4		unieq 3928	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
5	4	fveq2d 5679	. . . . . 6 ⊢ (𝑦 = 𝑥 → (inl‘∪ 𝑦) = (inl‘∪ 𝑥))
6	2, 3, 5	ifbieq12d 3653	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦)) = if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
7	6	cbvmptv 4211	. . . 4 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))
8		suceq 4528	. . . . 5 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
9	8	cbvmptv 4211	. . . 4 ⊢ (𝑦 ∈ ω ↦ suc 𝑦) = (𝑥 ∈ ω ↦ suc 𝑥)
10		eqid 2234	. . . 4 ⊢ case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o)) = case((𝑦 ∈ ω ↦ suc 𝑦), ( I ↾ 1o))
11	7, 9, 10	omp1eomlem 7398	. . 3 ⊢ (𝑦 ∈ ω ↦ if(𝑦 = ∅, (inr‘𝑦), (inl‘∪ 𝑦))):ω–1-1-onto→(ω ⊔ 1o)
12		f1oeng 7009	. . 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 7035	1 ⊢ (ω ⊔ 1o) ≈ ω

Syntax hints:   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ∅c0 3512  ifcif 3624  ∪ cuni 3919   class class class wbr 4114   ↦ cmpt 4176   I cid 4414  suc csuc 4491  ωcom 4717   ↾ cres 4756  –1-1-onto→wf1o 5356  ‘cfv 5357  1_oc1o 6653   ≈ cen 6986   ⊔ cdju 7341  inlcinl 7349  inrcinr 7350  casecdjucase 7387

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-er 6780  df-en 6989  df-dju 7342  df-inl 7351  df-inr 7352  df-case 7388

This theorem is referenced by:  difinfsn  7404  sbthom  16945