Theorem onintunirab 44432

Description: The intersection of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 29-Jan-2025.)

Assertion

Ref	Expression
onintunirab	⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem onintunirab

Step	Hyp	Ref	Expression
1		simp3 1156	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
2		ssint 4927	. . . . . . 7 ⊢ (𝑥 ⊆ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
3	1, 2	sylibr 237	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ⊆ ∩ 𝐴)
4		simp2 1155	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ∈ On)
5		oninton 7836	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On)
6	5	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → ∩ 𝐴 ∈ On)
7		onsssuc 6476	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ∩ 𝐴 ∈ On) → (𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ∈ suc ∩ 𝐴))
8	4, 6, 7	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → (𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ∈ suc ∩ 𝐴))
9	3, 8	mpbid 235	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ∈ suc ∩ 𝐴)
10	9	rabssdv 4002	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ suc ∩ 𝐴)
11		ssrab2 4008	. . . . . 6 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ On
12	11	a1i 11	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ On)
13		eloni 6393	. . . . . 6 ⊢ (∩ 𝐴 ∈ On → Ord ∩ 𝐴)
14	5, 13	syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → Ord ∩ 𝐴)
15		ordunisssuc 6492	. . . . 5 ⊢ (({𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ On ∧ Ord ∩ 𝐴) → (∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ ∩ 𝐴 ↔ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ suc ∩ 𝐴))
16	12, 14, 15	syl2anc 596	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → (∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ ∩ 𝐴 ↔ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ suc ∩ 𝐴))
17	10, 16	mpbird 260	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ ∩ 𝐴)
18		sseq1 3938	. . . . 5 ⊢ (𝑥 = ∩ 𝐴 → (𝑥 ⊆ 𝑦 ↔ ∩ 𝐴 ⊆ 𝑦))
19	18	ralbidv 3189	. . . 4 ⊢ (𝑥 = ∩ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦))
20		intss1 4926	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑦)
21	20	rgen 3082	. . . . 5 ⊢ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦
22	21	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦)
23	19, 5, 22	elrabd 3650	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
24		unissel 4903	. . 3 ⊢ ((∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ ∩ 𝐴 ∧ ∩ 𝐴 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∩ 𝐴)
25	17, 23, 24	syl2anc 596	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∩ 𝐴)
26	25	eqcomd 2769	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  {crab 3416   ⊆ wss 3900  ∅c0 4281  ∪ cuni 4870  ∩ cint 4910  Ord word 6382  Oncon0 6383  suc csuc 6385

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-tr 5217  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-ord 6386  df-on 6387  df-suc 6389

This theorem is used by:  oninfunirab  44442  oninfcl2  44443  oninfex2  44450