Theorem difex2 1951

Description: If the subtrahend of a class difference exists, then the minuend exists iff the difference exists.

Assertion

Ref	Expression
difex2	⊢ (B ∈ C → (A ∈ V ↔ (A ∖ B) ∈ V))

Proof of Theorem difex2

Step	Hyp	Ref	Expression
1		difexg 1703	. . 3 ⊢ (A ∈ V → (A ∖ B) ∈ V)
2	1	a1i 7	. 2 ⊢ (B ∈ C → (A ∈ V → (A ∖ B) ∈ V))
3		elisset 1354	. . . . . . . . 9 ⊢ (B ∈ C → B ∈ V)
4	3	anim1i 269	. . . . . . . 8 ⊢ ((B ∈ C ∧ (A ∖ B) ∈ V) → (B ∈ V ∧ (A ∖ B) ∈ V))
5	4	ancoms 334	. . . . . . 7 ⊢ (((A ∖ B) ∈ V ∧ B ∈ C) → (B ∈ V ∧ (A ∖ B) ∈ V))
6		unexb 1950	. . . . . . 7 ⊢ ((B ∈ V ∧ (A ∖ B) ∈ V) ↔ (B ∪ (A ∖ B)) ∈ V)
7	5, 6	sylib 173	. . . . . 6 ⊢ (((A ∖ B) ∈ V ∧ B ∈ C) → (B ∪ (A ∖ B)) ∈ V)
8		undif2 1762	. . . . . . 7 ⊢ (B ∪ (A ∖ B)) = (B ∪ A)
9	8	eleq1i 1152	. . . . . 6 ⊢ ((B ∪ (A ∖ B)) ∈ V ↔ (B ∪ A) ∈ V)
10	7, 9	sylib 173	. . . . 5 ⊢ (((A ∖ B) ∈ V ∧ B ∈ C) → (B ∪ A) ∈ V)
11		ssun2 1622	. . . . . 6 ⊢ A ⊆ (B ∪ A)
12		ssexg 1702	. . . . . 6 ⊢ ((B ∪ A) ∈ V → (A ⊆ (B ∪ A) → A ∈ V))
13	11, 12	mpi 44	. . . . 5 ⊢ ((B ∪ A) ∈ V → A ∈ V)
14	10, 13	syl 12	. . . 4 ⊢ (((A ∖ B) ∈ V ∧ B ∈ C) → A ∈ V)
15	14	exp 291	. . 3 ⊢ ((A ∖ B) ∈ V → (B ∈ C → A ∈ V))
16	15	com12 13	. 2 ⊢ (B ∈ C → ((A ∖ B) ∈ V → A ∈ V))
17	2, 16	impbid 397	1 ⊢ (B ∈ C → (A ∈ V ↔ (A ∖ B) ∈ V))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ⊆ wss 1487

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

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