Theorem hashinfuni 11138

Description: The ordinal size of an infinite set is ω. (Contributed by Jim Kingdon, 20-Feb-2022.)

Assertion

Ref	Expression
hashinfuni	⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)

Distinct variable group:   𝑦,𝐴

Proof of Theorem hashinfuni

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		omex 4714	. . . . . 6 ⊢ ω ∈ V
2	1	snid 3719	. . . . 5 ⊢ ω ∈ {ω}
3		elun2 3386	. . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω}))
4		breq1 4111	. . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴))
5	4	elrab3 2973	. . . . 5 ⊢ (ω ∈ (ω ∪ {ω}) → (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ↔ ω ≼ 𝐴))
6	2, 3, 5	mp2b 8	. . . 4 ⊢ (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ↔ ω ≼ 𝐴)
7	6	biimpri 133	. . 3 ⊢ (ω ≼ 𝐴 → ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
8		elrabi 2969	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω}))
9		elun 3359	. . . . . . 7 ⊢ (𝑧 ∈ (ω ∪ {ω}) ↔ (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
10	8, 9	sylib 122	. . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → (𝑧 ∈ ω ∨ 𝑧 ∈ {ω}))
11		ordom 4728	. . . . . . . 8 ⊢ Ord ω
12		ordelss 4499	. . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω)
13	11, 12	mpan 424	. . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω)
14		elsni 3706	. . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω)
15		eqimss 3291	. . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω)
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω)
17	13, 16	jaoi 724	. . . . . 6 ⊢ ((𝑧 ∈ ω ∨ 𝑧 ∈ {ω}) → 𝑧 ⊆ ω)
18	10, 17	syl 14	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ⊆ ω)
19	18	adantl 277	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) → 𝑧 ⊆ ω)
20	19	ralrimiva 2615	. . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω)
21		ssunieq 3946	. . 3 ⊢ ((ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∧ ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω) → ω = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
22	7, 20, 21	syl2anc 411	. 2 ⊢ (ω ≼ 𝐴 → ω = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
23	22	eqcomd 2238	1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2203  ∀wral 2520  {crab 2524   ∪ cun 3208   ⊆ wss 3210  {csn 3688  ∪ cuni 3913   class class class wbr 4108  Ord word 4482  ωcom 4711   ≼ cdom 6973

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-tr 4208  df-iord 4486  df-suc 4491  df-iom 4712

This theorem is referenced by:  hashinfom  11139