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Theorem difex2 7487
 Description: If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
difex2 (𝐵𝐶 → (𝐴 ∈ V ↔ (𝐴𝐵) ∈ V))

Proof of Theorem difex2
StepHypRef Expression
1 difexg 5201 . 2 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
2 ssun2 4080 . . . . 5 𝐴 ⊆ (𝐵𝐴)
3 uncom 4060 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴𝐵))
4 undif2 4376 . . . . . 6 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
53, 4eqtr2i 2782 . . . . 5 (𝐵𝐴) = ((𝐴𝐵) ∪ 𝐵)
62, 5sseqtri 3930 . . . 4 𝐴 ⊆ ((𝐴𝐵) ∪ 𝐵)
7 unexg 7476 . . . 4 (((𝐴𝐵) ∈ V ∧ 𝐵𝐶) → ((𝐴𝐵) ∪ 𝐵) ∈ V)
8 ssexg 5197 . . . 4 ((𝐴 ⊆ ((𝐴𝐵) ∪ 𝐵) ∧ ((𝐴𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
96, 7, 8sylancr 590 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐵𝐶) → 𝐴 ∈ V)
109expcom 417 . 2 (𝐵𝐶 → ((𝐴𝐵) ∈ V → 𝐴 ∈ V))
111, 10impbid2 229 1 (𝐵𝐶 → (𝐴 ∈ V ↔ (𝐴𝐵) ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  Vcvv 3409   ∖ cdif 3857   ∪ cun 3858   ⊆ wss 3860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-uni 4802 This theorem is referenced by:  elpwun  7496
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