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Theorem 0ss 3443
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 3409 . . 3 ¬ 𝑥 ∈ ∅
21pm2.21i 636 . 2 (𝑥 ∈ ∅ → 𝑥𝐴)
32ssriv 3142 1 ∅ ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2135  wss 3112  c0 3405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-dif 3114  df-in 3118  df-ss 3125  df-nul 3406
This theorem is referenced by:  ss0b  3444  ssdifeq0  3487  sssnr  3728  ssprr  3731  uni0  3811  int0el  3849  0disj  3974  disjx0  3976  tr0  4086  0elpw  4138  exmidsssn  4176  fr0  4324  elomssom  4577  rel0  4724  0ima  4959  fun0  5241  f0  5373  el2oss1o  6403  oaword1  6431  0domg  6795  nnnninf  7082  exmidfodomrlemim  7149  pw1on  7174  sum0  11319  prod0  11516  ennnfonelemj0  12297  ennnfonelemkh  12308  0opn  12571  baspartn  12615  0cld  12679  ntr0  12701  bdeq0  13610  bj-omtrans  13699  nninfsellemsuc  13753
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