Theorem bnj1418 34579

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 5144	. 2 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝑥 ↔ 𝑦𝑅𝑥))
2		df-bnj14 34228	. . 3 ⊢ pred(𝑥, 𝐴, 𝑅) = {𝑧 ∈ 𝐴 ∣ 𝑧𝑅𝑥}
3	2	bnj1538 34394	. 2 ⊢ (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝑅𝑥)
4	1, 3	vtoclga 3560	1 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)

Syntax hints:   → wi 4   ∈ wcel 2098   class class class wbr 5141   predc-bnj14 34227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-bnj14 34228

This theorem is referenced by:  bnj1417  34580  bnj1523  34610