Theorem hashinfuni 10774

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 4606	. . . . . 6 ⊢ ω ∈ V
2	1	snid 3637	. . . . 5 ⊢ ω ∈ {ω}
3		elun2 3317	. . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω}))
4		breq1 4020	. . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴))
5	4	elrab3 2908	. . . . 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 2904	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω}))
9		elun 3290	. . . . . . 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 4620	. . . . . . . 8 ⊢ Ord ω
12		ordelss 4393	. . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω)
13	11, 12	mpan 424	. . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω)
14		elsni 3624	. . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω)
15		eqimss 3223	. . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω)
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω)
17	13, 16	jaoi 717	. . . . . 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 2562	. . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω)
21		ssunieq 3856	. . 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 2194	1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 709   = wceq 1363   ∈ wcel 2159  ∀wral 2467  {crab 2471   ∪ cun 3141   ⊆ wss 3143  {csn 3606  ∪ cuni 3823   class class class wbr 4017  Ord word 4376  ωcom 4603   ≼ cdom 6756

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-iinf 4601

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-br 4018  df-tr 4116  df-iord 4380  df-suc 4385  df-iom 4604

This theorem is referenced by:  hashinfom  10775