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Theorem difex2 7781
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 5328 . 2 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
2 ssun2 4178 . . . . 5 𝐴 ⊆ (𝐵𝐴)
3 uncom 4157 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴𝐵))
4 undif2 4476 . . . . . 6 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
53, 4eqtr2i 2765 . . . . 5 (𝐵𝐴) = ((𝐴𝐵) ∪ 𝐵)
62, 5sseqtri 4031 . . . 4 𝐴 ⊆ ((𝐴𝐵) ∪ 𝐵)
7 unexg 7764 . . . 4 (((𝐴𝐵) ∈ V ∧ 𝐵𝐶) → ((𝐴𝐵) ∪ 𝐵) ∈ V)
8 ssexg 5322 . . . 4 ((𝐴 ⊆ ((𝐴𝐵) ∪ 𝐵) ∧ ((𝐴𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
96, 7, 8sylancr 587 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐵𝐶) → 𝐴 ∈ V)
109expcom 413 . 2 (𝐵𝐶 → ((𝐴𝐵) ∈ V → 𝐴 ∈ V))
111, 10impbid2 226 1 (𝐵𝐶 → (𝐴 ∈ V ↔ (𝐴𝐵) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2107  Vcvv 3479  cdif 3947  cun 3948  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-sn 4626  df-pr 4628  df-uni 4907
This theorem is referenced by:  elpwun  7790
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