Theorem bnj1418 31108

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 4656	. 2 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝑥 ↔ 𝑦𝑅𝑥))
2		df-bnj14 30755	. . 3 ⊢ pred(𝑥, 𝐴, 𝑅) = {𝑧 ∈ 𝐴 ∣ 𝑧𝑅𝑥}
3	2	bnj1538 30925	. 2 ⊢ (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝑅𝑥)
4	1, 3	vtoclga 3272	1 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)

Syntax hints:   → wi 4   ∈ wcel 1990   class class class wbr 4653   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-bnj14 30755

This theorem is referenced by:  bnj1417  31109  bnj1523  31139