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Theorem pssnel 3615
 Description: A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.)
Assertion
Ref Expression
pssnel (ABx(x B ¬ x A))
Distinct variable groups:   x,A   x,B

Proof of Theorem pssnel
StepHypRef Expression
1 pssdif 3612 . . 3 (AB → (B A) ≠ )
2 n0 3559 . . 3 ((B A) ≠ x x (B A))
31, 2sylib 188 . 2 (ABx x (B A))
4 eldif 3221 . . 3 (x (B A) ↔ (x B ¬ x A))
54exbii 1582 . 2 (x x (B A) ↔ x(x B ¬ x A))
63, 5sylib 188 1 (ABx(x B ¬ x A))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206   ⊊ wpss 3258  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-pss 3261  df-nul 3551 This theorem is referenced by: (None)
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