Theorem pwundifss 4263

Description: Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)

Assertion

Ref	Expression
pwundifss	⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Proof of Theorem pwundifss

Step	Hyp	Ref	Expression
1		undif1ss 3483	. 2 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) ⊆ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴)
2		pwunss 4261	. . . . 5 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		unss 3296	. . . . 5 ⊢ ((𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)) ↔ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵))
4	2, 3	mpbir 145	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵))
5	4	simpli 110	. . 3 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
6		ssequn2 3295	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵))
7	5, 6	mpbi 144	. 2 ⊢ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵)
8	1, 7	sseqtri 3176	1 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 103   = wceq 1343   ∖ cdif 3113   ∪ cun 3114   ⊆ wss 3116  𝒫 cpw 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by: (None)