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Theorem 0ss 3321
Description: The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
0ss ∅ ⊆ 𝐴

Proof of Theorem 0ss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 noel 3290 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 610 . 2 (𝑥 ∈ ∅ → 𝑥𝐴)
32ssriv 3029 1 ∅ ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1438  wss 2999  c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287
This theorem is referenced by:  ss0b  3322  ssdifeq0  3365  sssnr  3597  ssprr  3600  uni0  3680  int0el  3718  0disj  3842  disjx0  3844  tr0  3947  0elpw  3999  fr0  4178  elnn  4420  rel0  4562  0ima  4792  fun0  5072  f0  5201  oaword1  6232  0domg  6553  nnnninf  6806  exmidfodomrlemim  6827  sum0  10780  0opn  11603  bdeq0  11758  bj-omtrans  11851  el2oss1o  11887  nninfsellemsuc  11904
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