Theorem orduniss2 7680

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 3073	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)}
2		incom 4135	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On})
3		inab 4233	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)}
4		df-pw 4535	. . . . . . . 8 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
5	4	eqcomi 2747	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = 𝒫 𝐴
6		abid2 2882	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ On} = On
7	5, 6	ineq12i 4144	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On}) = (𝒫 𝐴 ∩ On)
8	2, 3, 7	3eqtr3i 2774	. . . . 5 ⊢ {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} = (𝒫 𝐴 ∩ On)
9	1, 8	eqtri 2766	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = (𝒫 𝐴 ∩ On)
10		ordpwsuc 7662	. . . 4 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
11	9, 10	eqtrid 2790	. . 3 ⊢ (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = suc 𝐴)
12	11	unieqd 4853	. 2 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = ∪ suc 𝐴)
13		ordunisuc 7679	. 2 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
14	12, 13	eqtrd 2778	1 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  {crab 3068   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  Ord word 6265  Oncon0 6266  suc csuc 6268

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-suc 6272

This theorem is referenced by: (None)