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	⊢ ∅ ⊆ 𝐴

Proof of Theorem 0ss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3255	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 585	. 2 ⊢ (𝑥 ∈ ∅ → 𝑥 ∈ 𝐴)
3	2	ssriv 2976	1 ⊢ ∅ ⊆ 𝐴

Syntax hints:   ∈ wcel 1409   ⊆ wss 2944  ∅c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038

This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-in 2951  df-ss 2958  df-nul 3252

This theorem is referenced by:  ss0b  3283  0pss  3292  npss0  3293  ssdifeq0  3332  sssnr  3551  ssprr  3554  uni0  3634  int0el  3672  0disj  3788  disjx0  3790  tr0  3892  0elpw  3944  fr0  4115  elnn  4355  rel0  4489  0ima  4712  fun0  4984  f0  5107  oaword1  6080  bdeq0  10353  bj-omtrans  10447