MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elpred Structured version   Visualization version   GIF version

Theorem elpred 5652
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)
Hypothesis
Ref Expression
elpred.1 𝑌 ∈ V
Assertion
Ref Expression
elpred (𝑋𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))

Proof of Theorem elpred
StepHypRef Expression
1 df-pred 5639 . . 3 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
21elin2 3779 . 2 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})))
3 elpred.1 . . . 4 𝑌 ∈ V
43eliniseg 5453 . . 3 (𝑋𝐷 → (𝑌 ∈ (𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋))
54anbi2d 739 . 2 (𝑋𝐷 → ((𝑌𝐴𝑌 ∈ (𝑅 “ {𝑋})) ↔ (𝑌𝐴𝑌𝑅𝑋)))
62, 5syl5bb 272 1 (𝑋𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  Vcvv 3186  {csn 4148   class class class wbr 4613  ccnv 5073  cima 5077  Predcpred 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639
This theorem is referenced by:  predpo  5657  setlikespec  5660  preddowncl  5666  wfrlem10  7369  preduz  12402  predfz  12405  wzel  31472  wzelOLD  31473  wsuclem  31474  wsuclemOLD  31475
  Copyright terms: Public domain W3C validator