Theorem onintunirab 42568

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 1136	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
2		ssint 4962	. . . . . . 7 ⊢ (𝑥 ⊆ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
3	1, 2	sylibr 233	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ⊆ ∩ 𝐴)
4		simp2 1135	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ∈ On)
5		oninton 7790	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On)
6	5	3ad2ant1 1131	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → ∩ 𝐴 ∈ On)
7		onsssuc 6453	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ∩ 𝐴 ∈ On) → (𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ∈ suc ∩ 𝐴))
8	4, 6, 7	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → (𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ∈ suc ∩ 𝐴))
9	3, 8	mpbid 231	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ∈ suc ∩ 𝐴)
10	9	rabssdv 4068	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ suc ∩ 𝐴)
11		ssrab2 4073	. . . . . 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 6373	. . . . . 6 ⊢ (∩ 𝐴 ∈ On → Ord ∩ 𝐴)
14	5, 13	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → Ord ∩ 𝐴)
15		ordunisssuc 6469	. . . . 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 583	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → (∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ ∩ 𝐴 ↔ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ suc ∩ 𝐴))
17	10, 16	mpbird 257	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ ∩ 𝐴)
18		sseq1 4003	. . . . 5 ⊢ (𝑥 = ∩ 𝐴 → (𝑥 ⊆ 𝑦 ↔ ∩ 𝐴 ⊆ 𝑦))
19	18	ralbidv 3172	. . . 4 ⊢ (𝑥 = ∩ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦))
20		intss1 4961	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑦)
21	20	rgen 3058	. . . . 5 ⊢ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦
22	21	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦)
23	19, 5, 22	elrabd 3682	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
24		unissel 4936	. . 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 583	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∩ 𝐴)
26	25	eqcomd 2733	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  {crab 3427   ⊆ wss 3944  ∅c0 4318  ∪ cuni 4903  ∩ cint 4944  Ord word 6362  Oncon0 6363  suc csuc 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-suc 6369

This theorem is referenced by:  oninfunirab  42578  oninfcl2  42579  oninfex2  42586