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Theorem orduniss2 7773
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 3411 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
2 incom 4166 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On})
3 inab 4264 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
4 df-pw 4567 . . . . . . . 8 𝒫 𝐴 = {𝑥𝑥𝐴}
54eqcomi 2746 . . . . . . 7 {𝑥𝑥𝐴} = 𝒫 𝐴
6 abid2 2876 . . . . . . 7 {𝑥𝑥 ∈ On} = On
75, 6ineq12i 4175 . . . . . 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 7755 . . . 4 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
119, 10eqtrid 2789 . . 3 (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
1211unieqd 4884 . 2 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
13 ordunisuc 7772 . 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 397   = wceq 1542  wcel 2107  {cab 2714  {crab 3410  cin 3914  wss 3915  𝒫 cpw 4565   cuni 4870  Ord word 6321  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by: (None)
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