Theorem nssne1 3328

Description: Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Assertion

Ref	Expression
nssne1	⊢ ((A ⊆ B ∧ ¬ A ⊆ C) → B ≠ C)

Proof of Theorem nssne1

Step	Hyp	Ref	Expression
1		sseq2 3294	. . . 4 ⊢ (B = C → (A ⊆ B ↔ A ⊆ C))
2	1	biimpcd 215	. . 3 ⊢ (A ⊆ B → (B = C → A ⊆ C))
3	2	necon3bd 2554	. 2 ⊢ (A ⊆ B → (¬ A ⊆ C → B ≠ C))
4	3	imp 418	1 ⊢ ((A ⊆ B ∧ ¬ A ⊆ C) → B ≠ C)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2517   ⊆ wss 3258

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-ss 3260

This theorem is referenced by: (None)