Theorem pred0 6282

Description: The predecessor class over ∅ is always ∅. (Contributed by Scott Fenton, 16-Apr-2011.) (Proof shortened by AV, 11-Jun-2021.)

Assertion

Ref	Expression
pred0	⊢ Pred(𝑅, ∅, 𝑋) = ∅

Proof of Theorem pred0

Step	Hyp	Ref	Expression
1		df-pred 6248	. 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋}))
2		0in 4347	. 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅
3	1, 2	eqtri 2754	1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅

Syntax hints:   = wceq 1541   ∩ cin 3901  ∅c0 4283  {csn 4576  ^◡ccnv 5615   “ cima 5619  Predcpred 6247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-in 3909  df-nul 4284  df-pred 6248

This theorem is referenced by: (None)