Theorem ssdifeq0 3632

Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)

Assertion

Ref	Expression
ssdifeq0	⊢ (A ⊆ (B ∖ A) ↔ A = ∅)

Proof of Theorem ssdifeq0

Step	Hyp	Ref	Expression
1		inidm 3464	. . 3 ⊢ (A ∩ A) = A
2		ssdifin0 3631	. . 3 ⊢ (A ⊆ (B ∖ A) → (A ∩ A) = ∅)
3	1, 2	syl5eqr 2399	. 2 ⊢ (A ⊆ (B ∖ A) → A = ∅)
4		0ss 3579	. . 3 ⊢ ∅ ⊆ (B ∖ ∅)
5		id 19	. . . 4 ⊢ (A = ∅ → A = ∅)
6		difeq2 3247	. . . 4 ⊢ (A = ∅ → (B ∖ A) = (B ∖ ∅))
7	5, 6	sseq12d 3300	. . 3 ⊢ (A = ∅ → (A ⊆ (B ∖ A) ↔ ∅ ⊆ (B ∖ ∅)))
8	4, 7	mpbiri 224	. 2 ⊢ (A = ∅ → A ⊆ (B ∖ A))
9	3, 8	impbii 180	1 ⊢ (A ⊆ (B ∖ A) ↔ A = ∅)

Syntax hints:   ↔ wb 176   = wceq 1642   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by: (None)