Theorem bnj213 34888

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 34695	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
2	1	ssrab3 4093	1 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Syntax hints:   ⊆ wss 3964   class class class wbr 5149   predc-bnj14 34694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3435  df-ss 3981  df-bnj14 34695

This theorem is referenced by:  bnj229  34890  bnj517  34891  bnj1128  34996  bnj1145  34999  bnj1137  35001  bnj1408  35042  bnj1417  35047  bnj1523  35077