Theorem orduniss2 7851

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 3416	. . . . 5 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)}
2		incom 4108	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On})
3		inab 4256	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ∈ On} ∩ {𝑥 ∣ 𝑥 ⊆ 𝐴}) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)}
4		df-pw 4561	. . . . . . . 8 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
5	4	eqcomi 2772	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = 𝒫 𝐴
6		abid2 2900	. . . . . . 7 ⊢ {𝑥 ∣ 𝑥 ∈ On} = On
7	5, 6	ineq12i 4117	. . . . . 6 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∩ {𝑥 ∣ 𝑥 ∈ On}) = (𝒫 𝐴 ∩ On)
8	2, 3, 7	3eqtr3i 2794	. . . . 5 ⊢ {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)} = (𝒫 𝐴 ∩ On)
9	1, 8	eqtri 2786	. . . 4 ⊢ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = (𝒫 𝐴 ∩ On)
10		ordpwsuc 7833	. . . 4 ⊢ (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
11	9, 10	eqtrid 2810	. . 3 ⊢ (Ord 𝐴 → {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = suc 𝐴)
12	11	unieqd 4882	. 2 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = ∪ suc 𝐴)
13		ordunisuc 7850	. 2 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
14	12, 13	eqtrd 2798	1 ⊢ (Ord 𝐴 → ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  {cab 2741  {crab 3415   ⊆ wss 3899   ∩ cin 4091  𝒫 cpw 4559  ∪ cuni 4869  Ord word 6367  Oncon0 6368  suc csuc 6370

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-tr 5215  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-ord 6371  df-on 6372  df-suc 6374

This theorem is used by: (None)