Theorem pssdifn0 4296

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 4294	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∖ 𝐴) = ∅)
2		eqss 3930	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	simplbi2 501	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐴 → 𝐴 = 𝐵))
4	1, 3	biimtrrid 244	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∖ 𝐴) = ∅ → 𝐴 = 𝐵))
5	4	necon3d 2955	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ≠ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅))
6	5	imp 407	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ≠ wne 2934   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-dif 3886  df-ss 3900  df-nul 4262

This theorem is referenced by:  pssdif  4297  tz7.7  6336  domdifsn  8988  inf3lem3  9542  isf32lem6  10271  fclscf  24008  flimfnfcls  24011  lebnumlem1  24946  lebnumlem2  24947  lebnumlem3  24948  ig1peu  26158  ig1pdvds  26163  qsidomlem2  33536  qsdrng  33580  divrngidl  38395