Theorem difex2 7739

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 5287	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐵) ∈ V)
2		ssun2 4145	. . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴)
3		uncom 4124	. . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴 ∖ 𝐵))
4		undif2 4443	. . . . . 6 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴)
5	3, 4	eqtr2i 2754	. . . . 5 ⊢ (𝐵 ∪ 𝐴) = ((𝐴 ∖ 𝐵) ∪ 𝐵)
6	2, 5	sseqtri 3998	. . . 4 ⊢ 𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵)
7		unexg 7722	. . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
8		ssexg 5281	. . . 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 413	. 2 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 ∖ 𝐵) ∈ V → 𝐴 ∈ V))
11	1, 10	impbid2 226	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-sn 4593  df-pr 4595  df-uni 4875

This theorem is referenced by:  elpwun  7748