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Theorem pssdifn0 3611
 Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
Assertion
Ref Expression
pssdifn0 ((A B AB) → (B A) ≠ )

Proof of Theorem pssdifn0
StepHypRef Expression
1 ssdif0 3609 . . . 4 (B A ↔ (B A) = )
2 eqss 3287 . . . . 5 (A = B ↔ (A B B A))
32simplbi2 608 . . . 4 (A B → (B AA = B))
41, 3syl5bir 209 . . 3 (A B → ((B A) = A = B))
54necon3d 2554 . 2 (A B → (AB → (B A) ≠ ))
65imp 418 1 ((A B AB) → (B A) ≠ )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516   ∖ cdif 3206   ⊆ wss 3257  ∅c0 3550 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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  pssdif  3612
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