Theorem pssdif 3613

Description: A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)

Assertion

Ref	Expression
pssdif	⊢ (A ⊊ B → (B ∖ A) ≠ ∅)

Proof of Theorem pssdif

Step	Hyp	Ref	Expression
1		df-pss 3262	. 2 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ A ≠ B))
2		pssdifn0 3612	. 2 ⊢ ((A ⊆ B ∧ A ≠ B) → (B ∖ A) ≠ ∅)
3	1, 2	sylbi 187	1 ⊢ (A ⊊ B → (B ∖ A) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 358   ≠ wne 2517   ∖ cdif 3207   ⊆ wss 3258   ⊊ wpss 3259  ∅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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-pss 3262  df-nul 3552

This theorem is referenced by:  pssnel  3616