Theorem ssdifeq0 4434

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 4197	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
2		ssdifin0 4433	. . 3 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → (𝐴 ∩ 𝐴) = ∅)
3	1, 2	syl5eqr 2872	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) → 𝐴 = ∅)
4		0ss 4352	. . 3 ⊢ ∅ ⊆ (𝐵 ∖ ∅)
5		id 22	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6		difeq2 4095	. . . 4 ⊢ (𝐴 = ∅ → (𝐵 ∖ 𝐴) = (𝐵 ∖ ∅))
7	5, 6	sseq12d 4002	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ ∅ ⊆ (𝐵 ∖ ∅)))
8	4, 7	mpbiri 260	. 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ (𝐵 ∖ 𝐴))
9	3, 8	impbii 211	1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 208   = wceq 1537   ∖ cdif 3935   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-in 3945  df-ss 3954  df-nul 4294

This theorem is referenced by:  disjdifprg  30327  measxun2  31471  measssd  31476  pmeasmono  31584