Theorem predel 6273

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 4154	. 2 ⊢ (𝑌 ∈ (𝐴 ∩ (◡𝑅 “ {𝑋})) → 𝑌 ∈ 𝐴)
2		df-pred 6253	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3	1, 2	eleq2s 2846	1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3904  {csn 4579  ^◡ccnv 5622   “ cima 5626  Predcpred 6252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-in 3912  df-pred 6253

This theorem is referenced by:  predtrss  6274  predpoirr  6285  predfrirr  6286  xpord2pred  8085