Theorem unixpss 5775

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 5774	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
2	1	unissi 4882	. . . 4 ⊢ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵)
3		unipw 5412	. . . 4 ⊢ ∪ 𝒫 𝒫 (𝐴 ∪ 𝐵) = 𝒫 (𝐴 ∪ 𝐵)
4	2, 3	sseqtri 3997	. . 3 ⊢ ∪ (𝐴 × 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
5	4	unissi 4882	. 2 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ ∪ 𝒫 (𝐴 ∪ 𝐵)
6		unipw 5412	. 2 ⊢ ∪ 𝒫 (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)
7	5, 6	sseqtri 3997	1 ⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵)

Syntax hints:   ∪ cun 3914   ⊆ wss 3916  𝒫 cpw 4565  ∪ cuni 4873   × cxp 5638

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-opab 5172  df-xp 5646  df-rel 5647

This theorem is referenced by:  relfld  6250  filnetlem3  36363