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Theorem orduniss2 7537
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 3144 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
2 incom 4175 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On})
3 inab 4268 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
4 df-pw 4537 . . . . . . . 8 𝒫 𝐴 = {𝑥𝑥𝐴}
54eqcomi 2827 . . . . . . 7 {𝑥𝑥𝐴} = 𝒫 𝐴
6 abid2 2954 . . . . . . 7 {𝑥𝑥 ∈ On} = On
75, 6ineq12i 4184 . . . . . 6 ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On}) = (𝒫 𝐴 ∩ On)
82, 3, 73eqtr3i 2849 . . . . 5 {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)} = (𝒫 𝐴 ∩ On)
91, 8eqtri 2841 . . . 4 {𝑥 ∈ On ∣ 𝑥𝐴} = (𝒫 𝐴 ∩ On)
10 ordpwsuc 7519 . . . 4 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
119, 10syl5eq 2865 . . 3 (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
1211unieqd 4840 . 2 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
13 ordunisuc 7536 . 2 (Ord 𝐴 suc 𝐴 = 𝐴)
1412, 13eqtrd 2853 1 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  {crab 3139  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830  Ord word 6183  Oncon0 6184  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190
This theorem is referenced by: (None)
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