Theorem predss 6312

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 6304	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inss1 4133	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
3	1, 2	eqsstri 3954	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

This proof depends on syntax axioms:   ⊆ wss 3895   ∩ cin 4089  {csn 4584  ^◡ccnv 5649   “ cima 5653  Predcpred 6303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-in 4091  df-pred 6304

This theorem is used by:  frpoins3xpg  8160  frpoins3xp3g  8161  xpord2pred  8165  xpord3pred  8172  fpr3g  8307  frrlem4  8311  frrlem13  8320  fpr1  8325  wfr3g  8341  ttrclselem1  9730  frmin  9757  frr3g  9764  frr1  9767  nummin  35775  wsuclem  36680