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Theorem difex2 7471
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 5222 . 2 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
2 ssun2 4146 . . . . 5 𝐴 ⊆ (𝐵𝐴)
3 uncom 4126 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴𝐵))
4 undif2 4421 . . . . . 6 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
53, 4eqtr2i 2842 . . . . 5 (𝐵𝐴) = ((𝐴𝐵) ∪ 𝐵)
62, 5sseqtri 4000 . . . 4 𝐴 ⊆ ((𝐴𝐵) ∪ 𝐵)
7 unexg 7461 . . . 4 (((𝐴𝐵) ∈ V ∧ 𝐵𝐶) → ((𝐴𝐵) ∪ 𝐵) ∈ V)
8 ssexg 5218 . . . 4 ((𝐴 ⊆ ((𝐴𝐵) ∪ 𝐵) ∧ ((𝐴𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
96, 7, 8sylancr 587 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐵𝐶) → 𝐴 ∈ V)
109expcom 414 . 2 (𝐵𝐶 → ((𝐴𝐵) ∈ V → 𝐴 ∈ V))
111, 10impbid2 227 1 (𝐵𝐶 → (𝐴 ∈ V ↔ (𝐴𝐵) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  Vcvv 3492  cdif 3930  cun 3931  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560  df-uni 4831
This theorem is referenced by:  elpwun  7480
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