Theorem pred0 6336

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 6300	. 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋}))
2		0in 4393	. 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅
3	1, 2	eqtri 2760	1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅

Syntax hints:   = wceq 1541   ∩ cin 3947  ∅c0 4322  {csn 4628  ^◡ccnv 5675   “ cima 5679  Predcpred 6299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-in 3955  df-nul 4323  df-pred 6300

This theorem is referenced by: (None)