Theorem bnj213 34866

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 34673	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
2	1	ssrab3 4041	1 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Syntax hints:   ⊆ wss 3911   class class class wbr 5102   predc-bnj14 34672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-ss 3928  df-bnj14 34673

This theorem is referenced by:  bnj229  34868  bnj517  34869  bnj1128  34974  bnj1145  34977  bnj1137  34979  bnj1408  35020  bnj1417  35025  bnj1523  35055