Theorem predss 6307

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 6299	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inss1 4197	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
3	1, 2	eqsstri 3991	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

Syntax hints:   ∩ cin 3912   ⊆ wss 3913  {csn 4591  ^◡ccnv 5658   “ cima 5662  Predcpred 6298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-in 3920  df-ss 3930  df-pred 6299

This theorem is referenced by:  frpoins3xpg  8132  frpoins3xp3g  8133  xpord2pred  8137  xpord3pred  8144  fpr3g  8278  frrlem4  8282  frrlem13  8291  fpr1  8296  wfr3g  8312  ttrclselem1  9690  frmin  9717  frr3g  9724  frr1  9727  nummin  35423  wsuclem  36210