Theorem predel 6344

Description: Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)

Assertion

Ref	Expression
predel	⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴)

Proof of Theorem predel

Step	Hyp	Ref	Expression
1		elinel1 4211	. 2 ⊢ (𝑌 ∈ (𝐴 ∩ (◡𝑅 “ {𝑋})) → 𝑌 ∈ 𝐴)
2		df-pred 6323	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3	1, 2	eleq2s 2857	1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2106   ∩ cin 3962  {csn 4631  ^◡ccnv 5688   “ cima 5692  Predcpred 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-pred 6323

This theorem is referenced by:  predtrss  6345  predpoirr  6356  predfrirr  6357  xpord2pred  8169