Theorem bnj213 35446

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 35254	. 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋}
2	1	ssrab3 4029	1 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴

This proof depends on syntax axioms:   ⊆ wss 3898   class class class wbr 5102   predc-bnj14 35253

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3915  df-bnj14 35254

This theorem is used by:  bnj229  35448  bnj517  35449  bnj1128  35554  bnj1145  35557  bnj1137  35559  bnj1408  35600  bnj1417  35605  bnj1523  35635