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Theorem 0el 3218
 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 2330 . 2 (∅ Ax A x = ∅)
2 eq0 3216 . . 3 (x = ∅ ↔ y ¬ y x)
32rexbii 2309 . 2 (x A x = ∅ ↔ x A y ¬ y x)
41, 3bitri 173 1 (∅ Ax A y ¬ y x)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  ∅c0 3201 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-dif 2897  df-nul 3202 This theorem is referenced by: (None)
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