Theorem bnj213 33722

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 33529	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
2	1	ssrab3 4076	1 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Syntax hints:   ⊆ wss 3944   class class class wbr 5141   predc-bnj14 33528

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3951  df-ss 3961  df-bnj14 33529

This theorem is referenced by:  bnj229  33724  bnj517  33725  bnj1128  33830  bnj1145  33833  bnj1137  33835  bnj1408  33876  bnj1417  33881  bnj1523  33911