Theorem difn0 4364

Description: If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)

Assertion

Ref	Expression
difn0	⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵)

Proof of Theorem difn0

Step	Hyp	Ref	Expression
1		eqimss 4040	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		ssdif0 4363	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	sylib 217	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅)
4	3	necon3i 2972	1 ⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ≠ wne 2939   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323

This theorem is referenced by:  disjdsct  32358  bj-2upln1upl  36371