Theorem pssdifn0 4321

Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)

Assertion

Ref	Expression
pssdifn0	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅)

Proof of Theorem pssdifn0

Step	Hyp	Ref	Expression
1		ssdif0 4319	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅)
2		eqss 3951	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	simplbi2 504	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐴 → 𝐴 = 𝐵))
4	1, 3	biimtrrid 245	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → 𝐴 = 𝐵))
5	4	necon3d 2978	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ≠ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅))
6	5	imp 410	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ≠ wne 2957   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-ss 3921  df-nul 4286

This theorem is referenced by:  pssdif  4322  tz7.7  6372  domdifsn  9032  inf3lem3  9585  isf32lem6  10315  fclscf  24082  flimfnfcls  24085  lebnumlem1  25020  lebnumlem2  25021  lebnumlem3  25022  ig1peu  26232  ig1pdvds  26237  qsidomlem2  33637  qsdrng  33682  dflringlem3  33689  dflring4  33691  divrngidl  38524