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Theorem 0el 3566
 Description: Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
Assertion
Ref Expression
0el ( Ax A y ¬ y x)
Distinct variable groups:   x,A   x,y
Allowed substitution hint:   A(y)

Proof of Theorem 0el
StepHypRef Expression
1 risset 2661 . 2 ( Ax A x = )
2 eq0 3564 . . 3 (x = y ¬ y x)
32rexbii 2639 . 2 (x A x = x A y ¬ y x)
41, 3bitri 240 1 ( Ax A y ¬ y x)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∅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-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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