Theorem bnj213 35215

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 35023	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
2	1	ssrab3 4042	1 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Syntax hints:   ⊆ wss 3911   class class class wbr 5111   predc-bnj14 35022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3423  df-ss 3928  df-bnj14 35023

This theorem is referenced by:  bnj229  35217  bnj517  35218  bnj1128  35323  bnj1145  35326  bnj1137  35328  bnj1408  35369  bnj1417  35374  bnj1523  35404