Theorem pssdifn0 3612

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 3610	. . . 4 ⊢ (B ⊆ A ↔ (B ∖ A) = ∅)
2		eqss 3288	. . . . 5 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
3	2	simplbi2 608	. . . 4 ⊢ (A ⊆ B → (B ⊆ A → A = B))
4	1, 3	syl5bir 209	. . 3 ⊢ (A ⊆ B → ((B ∖ A) = ∅ → A = B))
5	4	necon3d 2555	. 2 ⊢ (A ⊆ B → (A ≠ B → (B ∖ A) ≠ ∅))
6	5	imp 418	1 ⊢ ((A ⊆ B ∧ A ≠ B) → (B ∖ A) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2517   ∖ cdif 3207   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  pssdif  3613