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Theorem onuniintrab 43215
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 7821. (Contributed by RP, 28-Jan-2025.)
Assertion
Ref Expression
onuniintrab ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem onuniintrab
StepHypRef Expression
1 ssonuni 7799 . . 3 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
21impcom 407 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 ∈ On)
3 intmin 4973 . . 3 ( 𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = 𝐴)
4 unissb 4944 . . . . 5 ( 𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
54rabbii 3439 . . . 4 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
65inteqi 4955 . . 3 {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥}
73, 6eqtr3di 2790 . 2 ( 𝐴 ∈ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
82, 7syl 17 1 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963   cuni 4912   cint 4951  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onsupintrab  43220  onsupcl3  43222  onuniintrab2  43224
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