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Theorem orduniss2 7853
Description: The union of the ordinal subsets of an ordinal number is that number. (Contributed by NM, 30-Jan-2005.)
Assertion
Ref Expression
orduniss2 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem orduniss2
StepHypRef Expression
1 df-rab 3437 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
2 incom 4209 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On})
3 inab 4309 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
4 df-pw 4602 . . . . . . . 8 𝒫 𝐴 = {𝑥𝑥𝐴}
54eqcomi 2746 . . . . . . 7 {𝑥𝑥𝐴} = 𝒫 𝐴
6 abid2 2879 . . . . . . 7 {𝑥𝑥 ∈ On} = On
75, 6ineq12i 4218 . . . . . 6 ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On}) = (𝒫 𝐴 ∩ On)
82, 3, 73eqtr3i 2773 . . . . 5 {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)} = (𝒫 𝐴 ∩ On)
91, 8eqtri 2765 . . . 4 {𝑥 ∈ On ∣ 𝑥𝐴} = (𝒫 𝐴 ∩ On)
10 ordpwsuc 7835 . . . 4 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
119, 10eqtrid 2789 . . 3 (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
1211unieqd 4920 . 2 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
13 ordunisuc 7852 . 2 (Ord 𝐴 suc 𝐴 = 𝐴)
1412, 13eqtrd 2777 1 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  {crab 3436  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by: (None)
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