Theorem pssdif 4320

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 4017	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
2		pssdifn0 4319	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅)
3	1, 2	sylbi 220	1 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ≠ wne 2959   ⊆ wss 3900   ⊊ wpss 4016   ∖ cdif 4048  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-nul 4282

This theorem is used by:  pssnel  4427  pgpfac1lem5  20346  fundmpss  36747  dfon2lem6  36765