Theorem bnj213 35579

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 35387	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
2	1	ssrab3 4010	1 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

This proof depends on syntax axioms:   ⊆ wss 3900   class class class wbr 5106   predc-bnj14 35386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-ss 3901  df-bnj14 35387

This theorem is used by:  bnj229  35581  bnj517  35582  bnj1128  35687  bnj1145  35690  bnj1137  35692  bnj1408  35733  bnj1417  35738  bnj1523  35768