Theorem pred0 6343

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 6309	. 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋}))
2		0in 4357	. 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅
3	1, 2	eqtri 2789	1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3907  ∅c0 4289  {csn 4594  ^◡ccnv 5665   “ cima 5669  Predcpred 6308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-in 3915  df-nul 4290  df-pred 6309

This theorem is used by: (None)