MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difex2 Structured version   Visualization version   GIF version

Theorem difex2 7134
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 4960 . 2 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
2 ssun2 3920 . . . . 5 𝐴 ⊆ (𝐵𝐴)
3 uncom 3900 . . . . . 6 ((𝐴𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴𝐵))
4 undif2 4188 . . . . . 6 (𝐵 ∪ (𝐴𝐵)) = (𝐵𝐴)
53, 4eqtr2i 2783 . . . . 5 (𝐵𝐴) = ((𝐴𝐵) ∪ 𝐵)
62, 5sseqtri 3778 . . . 4 𝐴 ⊆ ((𝐴𝐵) ∪ 𝐵)
7 unexg 7124 . . . 4 (((𝐴𝐵) ∈ V ∧ 𝐵𝐶) → ((𝐴𝐵) ∪ 𝐵) ∈ V)
8 ssexg 4956 . . . 4 ((𝐴 ⊆ ((𝐴𝐵) ∪ 𝐵) ∧ ((𝐴𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
96, 7, 8sylancr 698 . . 3 (((𝐴𝐵) ∈ V ∧ 𝐵𝐶) → 𝐴 ∈ V)
109expcom 450 . 2 (𝐵𝐶 → ((𝐴𝐵) ∈ V → 𝐴 ∈ V))
111, 10impbid2 216 1 (𝐵𝐶 → (𝐴 ∈ V ↔ (𝐴𝐵) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  Vcvv 3340  cdif 3712  cun 3713  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324  df-uni 4589
This theorem is referenced by:  elpwun  7142
  Copyright terms: Public domain W3C validator