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Theorem pwundifss 4013
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 (AB) ∖ 𝒫 A) ∪ 𝒫 A) ⊆ 𝒫 (AB)

Proof of Theorem pwundifss
StepHypRef Expression
1 undif1ss 3292 . 2 ((𝒫 (AB) ∖ 𝒫 A) ∪ 𝒫 A) ⊆ (𝒫 (AB) ∪ 𝒫 A)
2 pwunss 4011 . . . . 5 (𝒫 A ∪ 𝒫 B) ⊆ 𝒫 (AB)
3 unss 3111 . . . . 5 ((𝒫 A ⊆ 𝒫 (AB) 𝒫 B ⊆ 𝒫 (AB)) ↔ (𝒫 A ∪ 𝒫 B) ⊆ 𝒫 (AB))
42, 3mpbir 134 . . . 4 (𝒫 A ⊆ 𝒫 (AB) 𝒫 B ⊆ 𝒫 (AB))
54simpli 104 . . 3 𝒫 A ⊆ 𝒫 (AB)
6 ssequn2 3110 . . 3 (𝒫 A ⊆ 𝒫 (AB) ↔ (𝒫 (AB) ∪ 𝒫 A) = 𝒫 (AB))
75, 6mpbi 133 . 2 (𝒫 (AB) ∪ 𝒫 A) = 𝒫 (AB)
81, 7sseqtri 2971 1 ((𝒫 (AB) ∖ 𝒫 A) ∪ 𝒫 A) ⊆ 𝒫 (AB)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  cdif 2908  cun 2909  wss 2911  𝒫 cpw 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by: (None)
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