Theorem pred0 6308

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 6274	. 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋}))
2		0in 4360	. 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅
3	1, 2	eqtri 2752	1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅

Syntax hints:   = wceq 1540   ∩ cin 3913  ∅c0 4296  {csn 4589  ^◡ccnv 5637   “ cima 5641  Predcpred 6273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-in 3921  df-nul 4297  df-pred 6274

This theorem is referenced by: (None)