Theorem bnj213 35279

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj213	⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Proof of Theorem bnj213

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bnj14 35087	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
2	1	ssrab3 4035	1 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

This proof depends on syntax axioms:   ⊆ wss 3904   class class class wbr 5108   predc-bnj14 35086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-ss 3921  df-bnj14 35087

This theorem is used by:  bnj229  35281  bnj517  35282  bnj1128  35387  bnj1145  35390  bnj1137  35392  bnj1408  35433  bnj1417  35438  bnj1523  35468