Theorem 0ss 3580

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.

Step	Hyp	Ref	Expression
1		noel 3555	. . 3 ⊢ ¬ x ∈ ∅
2	1	pm2.21i 123	. 2 ⊢ (x ∈ ∅ → x ∈ A)
3	2	ssriv 3278	1 ⊢ ∅ ⊆ A

Syntax hints:   ∈ wcel 1710   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  ss0b  3581  0pss  3589  npss0  3590  ssdifeq0  3633  pwpw0  3856  sssn  3865  sspr  3870  sstp  3871  pwsnALT  3883  uni0  3919  int0el  3958  iotassuni  4356  0ima  5015  dmxpss  5053  dmsnopss  5068  fun0  5155  f0  5249  fvmptss  5706  fvmptss2  5726  clos10  5888  mapsspm  6022  mapsspw  6023  map0e  6024  lec0cg  6199  0lt1c  6259  frecxp  6315