Theorem pssdif 4374

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 3982	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
2		pssdifn0 4373	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅)
3	1, 2	sylbi 217	1 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   ≠ wne 2937   ∖ cdif 3959   ⊆ wss 3962   ⊊ wpss 3963  ∅c0 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-v 3479  df-dif 3965  df-ss 3979  df-pss 3982  df-nul 4339

This theorem is referenced by:  pssnel  4476  pgpfac1lem5  20113  fundmpss  35747  dfon2lem6  35769