Theorem pred0 5710

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 5680	. 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋}))
2		0in 3969	. 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅
3	1, 2	eqtri 2644	1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅

Syntax hints:   = wceq 1483   ∩ cin 3573  ∅c0 3915  {csn 4177  ^◡ccnv 5113   “ cima 5117  Predcpred 5679

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-pred 5680

This theorem is referenced by:  trpred0  31736