Theorem predss 6311

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 6303	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inss1 4182	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
3	1, 2	eqsstri 3977	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

This proof depends on syntax axioms:   ∩ cin 3898   ⊆ wss 3899  {csn 4584  ^◡ccnv 5650   “ cima 5654  Predcpred 6302

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-in 3906  df-ss 3916  df-pred 6303

This theorem is used by:  frpoins3xpg  8150  frpoins3xp3g  8151  xpord2pred  8155  xpord3pred  8162  fpr3g  8296  frrlem4  8300  frrlem13  8309  fpr1  8314  wfr3g  8330  ttrclselem1  9719  frmin  9746  frr3g  9753  frr1  9756  nummin  35711  wsuclem  36567