Theorem onintunirab 42442

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 1137	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
2		ssint 4968	. . . . . . 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 1136	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ∈ On)
5		oninton 7787	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On)
6	5	3ad2ant1 1132	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → ∩ 𝐴 ∈ On)
7		onsssuc 6454	. . . . . . 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 4072	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ suc ∩ 𝐴)
11		ssrab2 4077	. . . . . 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 6374	. . . . . 6 ⊢ (∩ 𝐴 ∈ On → Ord ∩ 𝐴)
14	5, 13	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → Ord ∩ 𝐴)
15		ordunisssuc 6470	. . . . 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 4007	. . . . 5 ⊢ (𝑥 = ∩ 𝐴 → (𝑥 ⊆ 𝑦 ↔ ∩ 𝐴 ⊆ 𝑦))
19	18	ralbidv 3176	. . . 4 ⊢ (𝑥 = ∩ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦))
20		intss1 4967	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑦)
21	20	rgen 3062	. . . . 5 ⊢ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦
22	21	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦)
23	19, 5, 22	elrabd 3685	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
24		unissel 4942	. . 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 2737	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  {crab 3431   ⊆ wss 3948  ∅c0 4322  ∪ cuni 4908  ∩ cint 4950  Ord word 6363  Oncon0 6364  suc csuc 6366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-suc 6370

This theorem is referenced by:  oninfunirab  42452  oninfcl2  42453  oninfex2  42460