Theorem onuniintrab 43577

Description: The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Closed form of uniordint 7756. (Contributed by RP, 28-Jan-2025.)

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem onuniintrab

Step	Hyp	Ref	Expression
1		ssonuni 7735	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
2	1	impcom 407	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 ∈ On)
3		intmin 4925	. . 3 ⊢ (∪ 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∪ 𝐴)
4		unissb 4898	. . . . 5 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
5	4	rabbii 3406	. . . 4 ⊢ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}
6	5	inteqi 4908	. . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}
7	3, 6	eqtr3di 2787	. 2 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
8	2, 7	syl 17	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401   ⊆ wss 3903  ∪ cuni 4865  ∩ cint 4904  Oncon0 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329

This theorem is referenced by:  onsupintrab  43582  onsupcl3  43584  onuniintrab2  43586