Theorem predss 6314

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 6306	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inss1 4189	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
3	1, 2	eqsstri 3984	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

This proof depends on syntax axioms:   ∩ cin 3905   ⊆ wss 3906  {csn 4591  ^◡ccnv 5662   “ cima 5666  Predcpred 6305

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3913  df-ss 3923  df-pred 6306

This theorem is used by:  frpoins3xpg  8138  frpoins3xp3g  8139  xpord2pred  8143  xpord3pred  8150  fpr3g  8284  frrlem4  8288  frrlem13  8297  fpr1  8302  wfr3g  8318  ttrclselem1  9697  frmin  9724  frr3g  9731  frr1  9734  nummin  35518  wsuclem  36328