Theorem difn0 4323

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 4022	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
2		ssdif0 4322	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	sylib 220	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅)
4	3	necon3i 3048	1 ⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵)

Syntax hints:   → wi 4   = wceq 1533   ≠ wne 3016   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-in 3942  df-ss 3951  df-nul 4291

This theorem is referenced by:  disjdsct  30432  bj-2upln1upl  34331