Theorem pssdif 4324

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 3926	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
2		pssdifn0 4323	. 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 2960   ∖ cdif 3903   ⊆ wss 3906   ⊊ wpss 3907  ∅c0 4286

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3909  df-ss 3923  df-pss 3926  df-nul 4287

This theorem is used by:  pssnel  4431  pgpfac1lem5  20195  fundmpss  36296  dfon2lem6  36315