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Theorem elpwun 7348
 Description: Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)
Hypothesis
Ref Expression
eldifpw.1 𝐶 ∈ V
Assertion
Ref Expression
elpwun (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)

Proof of Theorem elpwun
StepHypRef Expression
1 elex 3455 . 2 (𝐴 ∈ 𝒫 (𝐵𝐶) → 𝐴 ∈ V)
2 elex 3455 . . 3 ((𝐴𝐶) ∈ 𝒫 𝐵 → (𝐴𝐶) ∈ V)
3 eldifpw.1 . . . 4 𝐶 ∈ V
4 difex2 7339 . . . 4 (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴𝐶) ∈ V))
53, 4ax-mp 5 . . 3 (𝐴 ∈ V ↔ (𝐴𝐶) ∈ V)
62, 5sylibr 235 . 2 ((𝐴𝐶) ∈ 𝒫 𝐵𝐴 ∈ V)
7 elpwg 4461 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ 𝐴 ⊆ (𝐵𝐶)))
8 difexg 5122 . . . . 5 (𝐴 ∈ V → (𝐴𝐶) ∈ V)
9 elpwg 4461 . . . . 5 ((𝐴𝐶) ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
108, 9syl 17 . . . 4 (𝐴 ∈ V → ((𝐴𝐶) ∈ 𝒫 𝐵 ↔ (𝐴𝐶) ⊆ 𝐵))
11 uncom 4050 . . . . . 6 (𝐵𝐶) = (𝐶𝐵)
1211sseq2i 3917 . . . . 5 (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴 ⊆ (𝐶𝐵))
13 ssundif 4347 . . . . 5 (𝐴 ⊆ (𝐶𝐵) ↔ (𝐴𝐶) ⊆ 𝐵)
1412, 13bitri 276 . . . 4 (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐶) ⊆ 𝐵)
1510, 14syl6rbbr 291 . . 3 (𝐴 ∈ V → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵))
167, 15bitrd 280 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵))
171, 6, 16pm5.21nii 380 1 (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∈ wcel 2081  Vcvv 3437   ∖ cdif 3856   ∪ cun 3857   ⊆ wss 3859  𝒫 cpw 4453 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-pw 4455  df-sn 4473  df-pr 4475  df-uni 4746 This theorem is referenced by:  pwfilem  8664  elrfi  38776  dssmapnvod  39851
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