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Theorem pwssunim 4174
Description: The power class of the union of two classes is a subset of the union of their power classes, if one class is a subclass of the other. One direction of Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by Jim Kingdon, 30-Sep-2018.)
Assertion
Ref Expression
pwssunim ((𝐴𝐵𝐵𝐴) → 𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))

Proof of Theorem pwssunim
StepHypRef Expression
1 ssequn2 3217 . . . . 5 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
2 pweq 3481 . . . . . 6 ((𝐴𝐵) = 𝐴 → 𝒫 (𝐴𝐵) = 𝒫 𝐴)
3 eqimss 3119 . . . . . 6 (𝒫 (𝐴𝐵) = 𝒫 𝐴 → 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐴)
42, 3syl 14 . . . . 5 ((𝐴𝐵) = 𝐴 → 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐴)
51, 4sylbi 120 . . . 4 (𝐵𝐴 → 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐴)
6 ssequn1 3214 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
7 pweq 3481 . . . . . 6 ((𝐴𝐵) = 𝐵 → 𝒫 (𝐴𝐵) = 𝒫 𝐵)
8 eqimss 3119 . . . . . 6 (𝒫 (𝐴𝐵) = 𝒫 𝐵 → 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐵)
97, 8syl 14 . . . . 5 ((𝐴𝐵) = 𝐵 → 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐵)
106, 9sylbi 120 . . . 4 (𝐴𝐵 → 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐵)
115, 10orim12i 731 . . 3 ((𝐵𝐴𝐴𝐵) → (𝒫 (𝐴𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐵))
1211orcoms 702 . 2 ((𝐴𝐵𝐵𝐴) → (𝒫 (𝐴𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐵))
13 ssun 3223 . 2 ((𝒫 (𝐴𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴𝐵) ⊆ 𝒫 𝐵) → 𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
1412, 13syl 14 1 ((𝐴𝐵𝐵𝐴) → 𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 680   = wceq 1314  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:  pwunim  4176
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