Theorem predss 6251

Description: The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

Ref	Expression
predss	⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

Proof of Theorem predss

Step	Hyp	Ref	Expression
1		df-pred 6243	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inss1 4182	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
3	1, 2	eqsstri 3976	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

Syntax hints:   ∩ cin 3896   ⊆ wss 3897  {csn 4571  ^◡ccnv 5610   “ cima 5614  Predcpred 6242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-ss 3914  df-pred 6243

This theorem is referenced by:  frpoins3xpg  8065  frpoins3xp3g  8066  xpord2pred  8070  xpord3pred  8077  fpr3g  8210  frrlem4  8214  frrlem13  8223  fpr1  8228  wfr3g  8244  ttrclselem1  9610  frmin  9637  frr3g  9644  frr1  9647  nummin  35096  wsuclem  35859