Theorem bnj213 35512

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 35320	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
2	1	ssrab3 4007	1 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

This proof depends on syntax axioms:   ⊆ wss 3895   class class class wbr 5102   predc-bnj14 35319

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3896  df-bnj14 35320

This theorem is used by:  bnj229  35514  bnj517  35515  bnj1128  35620  bnj1145  35623  bnj1137  35625  bnj1408  35666  bnj1417  35671  bnj1523  35701