Theorem bnj1418 31415

Description: Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1418	⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)

Proof of Theorem bnj1418

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4807	. 2 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝑥 ↔ 𝑦𝑅𝑥))
2		df-bnj14 31064	. . 3 ⊢ pred(𝑥, 𝐴, 𝑅) = {𝑧 ∈ 𝐴 ∣ 𝑧𝑅𝑥}
3	2	bnj1538 31232	. 2 ⊢ (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝑅𝑥)
4	1, 3	vtoclga 3412	1 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)

Syntax hints:   → wi 4   ∈ wcel 2139   class class class wbr 4804   predc-bnj14 31063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-bnj14 31064

This theorem is referenced by:  bnj1417  31416  bnj1523  31446