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Theorem orduniss2 6902
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 2904 . . . . 5 {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
2 incom 3766 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On})
3 inab 3853 . . . . . 6 ({𝑥𝑥 ∈ On} ∩ {𝑥𝑥𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
4 df-pw 4109 . . . . . . . 8 𝒫 𝐴 = {𝑥𝑥𝐴}
54eqcomi 2618 . . . . . . 7 {𝑥𝑥𝐴} = 𝒫 𝐴
6 abid2 2731 . . . . . . 7 {𝑥𝑥 ∈ On} = On
75, 6ineq12i 3773 . . . . . 6 ({𝑥𝑥𝐴} ∩ {𝑥𝑥 ∈ On}) = (𝒫 𝐴 ∩ On)
82, 3, 73eqtr3i 2639 . . . . 5 {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)} = (𝒫 𝐴 ∩ On)
91, 8eqtri 2631 . . . 4 {𝑥 ∈ On ∣ 𝑥𝐴} = (𝒫 𝐴 ∩ On)
10 ordpwsuc 6884 . . . 4 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
119, 10syl5eq 2655 . . 3 (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
1211unieqd 4376 . 2 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = suc 𝐴)
13 ordunisuc 6901 . 2 (Ord 𝐴 suc 𝐴 = 𝐴)
1412, 13eqtrd 2643 1 (Ord 𝐴 {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {cab 2595  {crab 2899  cin 3538  wss 3539  𝒫 cpw 4107   cuni 4366  Ord word 5624  Oncon0 5625  suc csuc 5627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4938  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-ord 5628  df-on 5629  df-suc 5631
This theorem is referenced by: (None)
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