Theorem ssdifeq0 3491

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 3331	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
2		ssdifin0 3490	. . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅)
3	1, 2	eqtr3id 2213	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅)
4		0ss 3447	. . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅)
5		id 19	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6		difeq2 3234	. . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅))
7	5, 6	sseq12d 3173	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅)))
8	4, 7	mpbiri 167	. 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴))
9	3, 8	impbii 125	1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 104   = wceq 1343   ∖ cdif 3113   ∩ cin 3115   ⊆ wss 3116  ∅c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410

This theorem is referenced by: (None)