Theorem pwundifss 4320

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 3525	. 2 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) ⊆ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴)
2		pwunss 4318	. . . . 5 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		unss 3337	. . . . 5 ⊢ ((𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)) ↔ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵))
4	2, 3	mpbir 146	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵))
5	4	simpli 111	. . 3 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
6		ssequn2 3336	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵))
7	5, 6	mpbi 145	. 2 ⊢ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵)
8	1, 7	sseqtri 3217	1 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1364   ∖ cdif 3154   ∪ cun 3155   ⊆ wss 3157  𝒫 cpw 3605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607

This theorem is referenced by: (None)