Theorem ssindif0 3605

Description: Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
ssindif0	⊢ (A ⊆ B ↔ (A ∩ (V ∖ B)) = ∅)

Proof of Theorem ssindif0

Step	Hyp	Ref	Expression
1		disj2 3599	. 2 ⊢ ((A ∩ (V ∖ B)) = ∅ ↔ A ⊆ (V ∖ (V ∖ B)))
2		ddif 3399	. . 3 ⊢ (V ∖ (V ∖ B)) = B
3	2	sseq2i 3297	. 2 ⊢ (A ⊆ (V ∖ (V ∖ B)) ↔ A ⊆ B)
4	1, 3	bitr2i 241	1 ⊢ (A ⊆ B ↔ (A ∩ (V ∖ B)) = ∅)

Syntax hints:   ↔ wb 176   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551

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 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)