Theorem hashinfuni 10759

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 4594	. . . . . 6 ⊢ ω ∈ V
2	1	snid 3625	. . . . 5 ⊢ ω ∈ {ω}
3		elun2 3305	. . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω}))
4		breq1 4008	. . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴))
5	4	elrab3 2896	. . . . 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 2892	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω}))
9		elun 3278	. . . . . . 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 4608	. . . . . . . 8 ⊢ Ord ω
12		ordelss 4381	. . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω)
13	11, 12	mpan 424	. . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω)
14		elsni 3612	. . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω)
15		eqimss 3211	. . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω)
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω)
17	13, 16	jaoi 716	. . . . . 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 2550	. . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω)
21		ssunieq 3844	. . 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 2183	1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω)

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∀wral 2455  {crab 2459   ∪ cun 3129   ⊆ wss 3131  {csn 3594  ∪ cuni 3811   class class class wbr 4005  Ord word 4364  ωcom 4591   ≼ cdom 6741

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-tr 4104  df-iord 4368  df-suc 4373  df-iom 4592

This theorem is referenced by:  hashinfom  10760