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Theorem pwunss 4173
Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
pwunss (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)

Proof of Theorem pwunss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssun 3223 . . 3 ((𝑥𝐴𝑥𝐵) → 𝑥 ⊆ (𝐴𝐵))
2 elun 3185 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
3 vex 2661 . . . . . 6 𝑥 ∈ V
43elpw 3484 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53elpw 3484 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
64, 5orbi12i 736 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
72, 6bitri 183 . . 3 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
83elpw 3484 . . 3 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
91, 7, 83imtr4i 200 . 2 (𝑥 ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) → 𝑥 ∈ 𝒫 (𝐴𝐵))
109ssriv 3069 1 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wo 680  wcel 1463  cun 3037  wss 3039  𝒫 cpw 3478
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480
This theorem is referenced by:  pwundifss  4175  pwunim  4176
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