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Theorem orduniss2 7404
 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 3114 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
2 incom 4099 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On})
3 inab 4191 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
4 df-pw 4455 . . . . . . . 8 𝒫 𝐴 = {𝑥𝑥𝐴}
54eqcomi 2804 . . . . . . 7 {𝑥𝑥𝐴} = 𝒫 𝐴
6 abid2 2926 . . . . . . 7 {𝑥𝑥 ∈ On} = On
75, 6ineq12i 4107 . . . . . 6 ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On}) = (𝒫 𝐴 ∩ On)
82, 3, 73eqtr3i 2827 . . . . 5 {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)} = (𝒫 𝐴 ∩ On)
91, 8eqtri 2819 . . . 4 {𝑥 ∈ On ∣ 𝑥𝐴} = (𝒫 𝐴 ∩ On)
10 ordpwsuc 7386 . . . 4 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
119, 10syl5eq 2843 . . 3 (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
1211unieqd 4755 . 2 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
13 ordunisuc 7403 . 2 (Ord 𝐴 suc 𝐴 = 𝐴)
1412, 13eqtrd 2831 1 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  {cab 2775  {crab 3109   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4453  ∪ cuni 4745  Ord word 6065  Oncon0 6066  suc csuc 6068 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-tr 5064  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-ord 6069  df-on 6070  df-suc 6072 This theorem is referenced by: (None)
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