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Theorem 0ss 3255
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  |-  (/)  C_  A

Proof of Theorem 0ss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 noel 3228 . . 3  |-  -.  x  e.  (/)
21pm2.21i 575 . 2  |-  ( x  e.  (/)  ->  x  e.  A )
32ssriv 2949 1  |-  (/)  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1393    C_ wss 2917   (/)c0 3224
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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225
This theorem is referenced by:  ss0b  3256  0pss  3265  npss0  3266  ssdifeq0  3305  sssnr  3524  ssprr  3527  uni0  3607  int0el  3645  0disj  3761  disjx0  3763  tr0  3865  0elpw  3917  fr0  4088  elnn  4328  rel0  4462  0ima  4685  fun0  4957  f0  5080  oaword1  6050  bdeq0  9987  bj-omtrans  10081
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