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 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 5654   “ cima 5658  Predcpred 6298

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 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 3906  df-ss 3916  df-pred 6299

This theorem is used by:  frpoins3xpg  8138  frpoins3xp3g  8139  xpord2pred  8143  xpord3pred  8150  fpr3g  8284  frrlem4  8288  frrlem13  8297  fpr1  8302  wfr3g  8318  ttrclselem1  9704  frmin  9731  frr3g  9738  frr1  9741  nummin  35598  wsuclem  36402