Theorem predasetex 6166

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

Step	Hyp	Ref	Expression
1		df-pred 6151	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		predasetex.1	. . 3 ⊢ 𝐴 ∈ V
3	2	inex1 5224	. 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V
4	1, 3	eqeltri 2912	1 ⊢ Pred(𝑅, 𝐴, 𝑋) ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3497   ∩ cin 3938  {csn 4570  ^◡ccnv 5557   “ cima 5561  Predcpred 6150

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-in 3946  df-pred 6151

This theorem is referenced by: (None)