Theorem uniordint 7798

Description: The union of a set of ordinals is equal to the intersection of its upper bounds. Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)

Hypothesis

Ref	Expression
uniordint.1	⊢ 𝐴 ∈ V

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem uniordint

Step	Hyp	Ref	Expression
1		uniordint.1	. . 3 ⊢ 𝐴 ∈ V
2	1	ssonunii 7778	. 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On)
3		intmin 4927	. . 3 ⊢ (∪ 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∪ 𝐴)
4		unissb 4900	. . . . 5 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
5	4	rabbii 3417	. . . 4 ⊢ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}
6	5	inteqi 4910	. . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}
7	3, 6	eqtr3di 2810	. 2 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
8	2, 7	syl 18	1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3898  ∪ cuni 4866  ∩ cint 4906  Oncon0 6351

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7734

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6354  df-on 6355

This theorem is used by: (None)