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Theorem predasetex 5882
Description: The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.)
Hypothesis
Ref Expression
predasetex.1 𝐴 ∈ V
Assertion
Ref Expression
predasetex Pred(𝑅, 𝐴, 𝑋) ∈ V

Proof of Theorem predasetex
StepHypRef Expression
1 df-pred 5867 . 2 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
2 predasetex.1 . . 3 𝐴 ∈ V
32inex1 4962 . 2 (𝐴 ∩ (𝑅 “ {𝑋})) ∈ V
41, 3eqeltri 2840 1 Pred(𝑅, 𝐴, 𝑋) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2155  Vcvv 3350  cin 3733  {csn 4336  ccnv 5278  cima 5282  Predcpred 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743  ax-sep 4943
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3741  df-pred 5867
This theorem is referenced by: (None)
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