Theorem difn0 4315

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 3989	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		ssdif0 4314	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	sylib 221	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅)
4	3	necon3i 2987	1 ⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2955   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279

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 2147  ax-9 2155  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 3902  df-ss 3916  df-nul 4280

This theorem is used by:  disjdsct  33175  bj-2upln1upl  37768