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Theorem prtex 38346
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 38345 . . 3 (Prt 𝐴 Er 𝐴)
3 erexb 8743 . . 3 ( Er 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
42, 3syl 17 . 2 (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
5 uniexb 7760 . 2 (𝐴 ∈ V ↔ 𝐴 ∈ V)
64, 5bitr4di 289 1 (Prt 𝐴 → ( ∈ V ↔ 𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wrex 3066  Vcvv 3470   cuni 4903  {copab 5204   Er wer 8715  Prt wprt 38337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-er 8718  df-prt 38338
This theorem is referenced by: (None)
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