Theorem predss 6338

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 6330	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		inss1 4136	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
3	1, 2	eqsstri 3959	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴

This proof depends on syntax axioms:   ⊆ wss 3900   ∩ cin 4092  {csn 4588  ^◡ccnv 5657   “ cima 5661  Predcpred 6329

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-in 4094  df-pred 6330

This theorem is used by:  frpoins3xpg  8194  frpoins3xp3g  8195  xpord2pred  8199  xpord3pred  8206  fpr3g  8341  frrlem4  8345  frrlem13  8354  fpr1  8359  wfr3g  8375  ttrclselem1  9763  frmin  9790  frr3g  9797  frr1  9800  nummin  35847  wsuclem  36802