Theorem 0ss 3282

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.

Step	Hyp	Ref	Expression
1		noel 3255	. . 3 |- -. x e. (/)
2	1	pm2.21i 607	. 2 |- ( x e. (/) -> x e. A )
3	2	ssriv 3003	1 |- (/) C_ A

Syntax hints:   e. wcel 1433   C_ wss 2973   (/)c0 3251

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252

This theorem is referenced by:  ss0b  3283  ssdifeq0  3325  sssnr  3545  ssprr  3548  uni0  3628  int0el  3666  0disj  3782  disjx0  3784  tr0  3886  0elpw  3938  fr0  4106  elnn  4346  rel0  4480  0ima  4705  fun0  4977  f0  5100  oaword1  6073  bdeq0  10658  bj-omtrans  10751