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Theorem eq0 3564
 Description: The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
Assertion
Ref Expression
eq0 (A = x ¬ x A)
Distinct variable group:   x,A

Proof of Theorem eq0
StepHypRef Expression
1 neq0 3560 . . 3 A = x x A)
2 df-ex 1542 . . 3 (x x A ↔ ¬ x ¬ x A)
31, 2bitri 240 . 2 A = ↔ ¬ x ¬ x A)
43con4bii 288 1 (A = x ¬ x A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∅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-nul 3551 This theorem is referenced by:  0el  3566  disj  3591  ssdif0  3609  difin0ss  3616  inssdif0  3617  ralf0  3656  addcnul1  4452  dm0  4918  dmeq0  4922  co01  5093  clos1nrel  5886  ncprc  6124
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