Theorem hashinfuni 11216

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 4740	. . . . . 6 ⊢ ω ∈ V
2	1	snid 3740	. . . . 5 ⊢ ω ∈ {ω}
3		elun2 3397	. . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω}))
4		breq1 4133	. . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴))
5	4	elrab3 2983	. . . . 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 2979	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω}))
9		elun 3370	. . . . . . 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 4754	. . . . . . . 8 ⊢ Ord ω
12		ordelss 4524	. . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω)
13	11, 12	mpan 428	. . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω)
14		elsni 3727	. . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω)
15		eqimss 3302	. . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω)
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω)
17	13, 16	jaoi 728	. . . . . 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 2623	. . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω)
21		ssunieq 3968	. . 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 415	. 2 ⊢ (ω ≼ 𝐴 → ω = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴})
23	22	eqcomd 2244	1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ∪ cun 3218   ⊆ wss 3220  {csn 3709  ∪ cuni 3935   class class class wbr 4130  Ord word 4507  ωcom 4737   ≼ cdom 7021

This proof depends on 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-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-tr 4230  df-iord 4511  df-suc 4516  df-iom 4738

This theorem is used by:  hashinfom  11217