Theorem onsupcl3 44235

Description: The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)

Assertion

Ref	Expression
onsupcl3	⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ On)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem onsupcl3

Step	Hyp	Ref	Expression
1		onuniintrab 44228	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
2		ssonuni 7794	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
3	2	impcom 413	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 ∈ On)
4	1, 3	eqeltrrd 2861	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ On)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895  ∪ cuni 4866  ∩ cint 4906  Oncon0 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  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 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366

This theorem is used by:  onsupex3  44236