Theorem predel 6133

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 4122	. 2 ⊢ (𝑌 ∈ (𝐴 ∩ (◡𝑅 “ {𝑋})) → 𝑌 ∈ 𝐴)
2		df-pred 6116	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3	1, 2	eleq2s 2908	1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2111   ∩ cin 3880  {csn 4525  ^◡ccnv 5518   “ cima 5522  Predcpred 6115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-pred 6116

This theorem is referenced by:  predpo  6134  predpoirr  6144  predfrirr  6145  dftrpred3g  33185