Theorem difss 3394

Description: Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)

Assertion

Ref	Expression
difss	⊢ (A ∖ B) ⊆ A

Proof of Theorem difss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 3389	. 2 ⊢ (x ∈ (A ∖ B) → x ∈ A)
2	1	ssriv 3278	1 ⊢ (A ∖ B) ⊆ A

Syntax hints:   ∖ cdif 3207   ⊆ wss 3258

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

This theorem is referenced by:  difssd  3395  difss2  3396  ssdifss  3398  disj4  3600  0dif  3622  uneqdifeq  3639  difsnpss  3852  unidif  3924  iunxdif2  4015  imagekrelk  4274  nnsucelr  4429  sfinltfin  4536  vfinncvntsp  4550  vfinspsslem1  4551  vfinncsp  4555  resdif  5307  sbthlem1  6204