Theorem difn0 4390

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 4067	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		ssdif0 4389	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	sylib 218	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅)
4	3	necon3i 2979	1 ⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ≠ wne 2946   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353

This theorem is referenced by:  disjdsct  32716  bj-2upln1upl  36992