Theorem unixpss 5820

Description: The double class union of a Cartesian product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
unixpss	⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵)

Proof of Theorem unixpss

Step	Hyp	Ref	Expression
1		xpsspw 5819	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
2	1	unissi 4916	. . . 4 ⊢ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵)
3		unipw 5455	. . . 4 ⊢ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) = 𝒫 (𝐴 ∪ 𝐵)
4	2, 3	sseqtri 4032	. . 3 ⊢ ∪ (𝐴 × 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
5	4	unissi 4916	. 2 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 (𝐴 ∪ 𝐵)
6		unipw 5455	. 2 ⊢ ∪ 𝒫 (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)
7	5, 6	sseqtri 4032	1 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∪ cun 3949   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907   × cxp 5683

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-opab 5206  df-xp 5691  df-rel 5692

This theorem is referenced by:  relfld  6295  filnetlem3  36381