Theorem orduniss2 7811

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 ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem orduniss2

Step	Hyp	Ref	Expression
1		df-rab 3409	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)}
2		incom 4175	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On})
3		inab 4275	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)}
4		df-pw 4568	. . . . . . . 8 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
5	4	eqcomi 2739	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = 𝒫 𝐴
6		abid2 2866	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ On} = On
7	5, 6	ineq12i 4184	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On}) = (𝒫 𝐴 ∩ On)
8	2, 3, 7	3eqtr3i 2761	. . . . 5 ⊢ {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} = (𝒫 𝐴 ∩ On)
9	1, 8	eqtri 2753	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = (𝒫 𝐴 ∩ On)
10		ordpwsuc 7793	. . . 4 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
11	9, 10	eqtrid 2777	. . 3 ⊢ (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = suc 𝐴)
12	11	unieqd 4887	. 2 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = ∪ suc 𝐴)
13		ordunisuc 7810	. 2 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
14	12, 13	eqtrd 2765	1 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  {crab 3408   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  Ord word 6334  Oncon0 6335  suc csuc 6337

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339  df-suc 6341

This theorem is referenced by: (None)