Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtex Structured version   Visualization version   GIF version

Theorem prtex 33984
Description: The equivalence relation generated by a partition is a set if and only if the partition itself is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
Assertion
Ref Expression
prtex (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
Distinct variable group:   𝑥,𝑢,𝑦,𝐴
Allowed substitution hints:   (𝑥,𝑦,𝑢)

Proof of Theorem prtex
StepHypRef Expression
1 prtlem18.1 . . . 4 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
21prter1 33983 . . 3 (Prt 𝐴 Er 𝐴)
3 erexb 7752 . . 3 ( Er 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
42, 3syl 17 . 2 (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
5 uniexb 6958 . 2 (𝐴 ∈ V ↔ 𝐴 ∈ V)
64, 5syl6bbr 278 1 (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wrex 2910  Vcvv 3195   cuni 4427  {copab 4703   Er wer 7724  Prt wprt 33975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-er 7727  df-prt 33976
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator