Theorem onintunirab 43469

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 1138	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
2		ssint 4919	. . . . . . 7 ⊢ (𝑥 ⊆ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦)
3	1, 2	sylibr 234	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ⊆ ∩ 𝐴)
4		simp2 1137	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ∈ On)
5		oninton 7740	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On)
6	5	3ad2ant1 1133	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → ∩ 𝐴 ∈ On)
7		onsssuc 6409	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ∩ 𝐴 ∈ On) → (𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ∈ suc ∩ 𝐴))
8	4, 6, 7	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → (𝑥 ⊆ ∩ 𝐴 ↔ 𝑥 ∈ suc ∩ 𝐴))
9	3, 8	mpbid 232	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝑥 ∈ suc ∩ 𝐴)
10	9	rabssdv 4026	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ⊆ suc ∩ 𝐴)
11		ssrab2 4032	. . . . . 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 6327	. . . . . 6 ⊢ (∩ 𝐴 ∈ On → Ord ∩ 𝐴)
14	5, 13	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → Ord ∩ 𝐴)
15		ordunisssuc 6425	. . . . 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 584	. . . 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 3959	. . . . 5 ⊢ (𝑥 = ∩ 𝐴 → (𝑥 ⊆ 𝑦 ↔ ∩ 𝐴 ⊆ 𝑦))
19	18	ralbidv 3159	. . . 4 ⊢ (𝑥 = ∩ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦))
20		intss1 4918	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑦)
21	20	rgen 3053	. . . . 5 ⊢ ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦
22	21	a1i 11	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∀𝑦 ∈ 𝐴 ∩ 𝐴 ⊆ 𝑦)
23	19, 5, 22	elrabd 3648	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
24		unissel 4895	. . 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 584	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∩ 𝐴)
26	25	eqcomd 2742	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  {crab 3399   ⊆ wss 3901  ∅c0 4285  ∪ cuni 4863  ∩ cint 4902  Ord word 6316  Oncon0 6317  suc csuc 6319

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by:  oninfunirab  43479  oninfcl2  43480  oninfex2  43487