Theorem pssdif 4332

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 3934	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
2		pssdifn0 4331	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅)
3	1, 2	sylbi 217	1 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   ≠ wne 2925   ∖ cdif 3911   ⊆ wss 3914   ⊊ wpss 3915  ∅c0 4296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3449  df-dif 3917  df-ss 3931  df-pss 3934  df-nul 4297

This theorem is referenced by:  pssnel  4434  pgpfac1lem5  20011  fundmpss  35754  dfon2lem6  35776