Theorem ssdifeq0 4051

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 3822	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
2		ssdifin0 4050	. . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅)
3	1, 2	syl5eqr 2670	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅)
4		0ss 3972	. . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅)
5		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6		difeq2 3722	. . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅))
7	5, 6	sseq12d 3634	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅)))
8	4, 7	mpbiri 248	. 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴))
9	3, 8	impbii 199	1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 196   = wceq 1483   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  disjdifprg  29388  measxun2  30273  measssd  30278  pmeasmono  30386