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Theorem 0ss 3299
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 3272 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 608 . 2 (𝑥 ∈ ∅ → 𝑥𝐴)
32ssriv 3013 1 ∅ ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1434  wss 2983  c0 3268
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-dif 2985  df-in 2989  df-ss 2996  df-nul 3269
This theorem is referenced by:  ss0b  3300  ssdifeq0  3342  sssnr  3566  ssprr  3569  uni0  3649  int0el  3687  0disj  3803  disjx0  3805  tr0  3907  0elpw  3959  fr0  4135  elnn  4375  rel0  4511  0ima  4736  fun0  5009  f0  5132  oaword1  6137  bdeq0  10950  bj-omtrans  11043
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