Theorem difex2 7746

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	⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V))

Proof of Theorem difex2

Step	Hyp	Ref	Expression
1		difexg 5327	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐵) ∈ V)
2		ssun2 4173	. . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴)
3		uncom 4153	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴 ∖ 𝐵))
4		undif2 4476	. . . . . 6 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴)
5	3, 4	eqtr2i 2761	. . . . 5 ⊢ (𝐵 ∪ 𝐴) = ((𝐴 ∖ 𝐵) ∪ 𝐵)
6	2, 5	sseqtri 4018	. . . 4 ⊢ 𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵)
7		unexg 7735	. . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
8		ssexg 5323	. . . 4 ⊢ ((𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
9	6, 7, 8	sylancr 587	. . 3 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
10	9	expcom 414	. 2 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 ∖ 𝐵) ∈ V → 𝐴 ∈ V))
11	1, 10	impbid2 225	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  Vcvv 3474   ∖ cdif 3945   ∪ cun 3946   ⊆ wss 3948

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909

This theorem is referenced by:  elpwun  7755