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

Theorem dprddomprc 18320
 Description: A family of subgroups indexed by a proper class cannot be a family of subgroups for an internal direct product. (Contributed by AV, 13-Jul-2019.)
Assertion
Ref Expression
dprddomprc (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)

Proof of Theorem dprddomprc
StepHypRef Expression
1 df-nel 2894 . . 3 (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
2 dmexg 7044 . . . 4 (𝑆 ∈ V → dom 𝑆 ∈ V)
32con3i 150 . . 3 (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V)
41, 3sylbi 207 . 2 (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V)
5 reldmdprd 18317 . . 3 Rel dom DProd
65brrelex2i 5119 . 2 (𝐺dom DProd 𝑆𝑆 ∈ V)
74, 6nsyl 135 1 (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1987   ∉ wnel 2893  Vcvv 3186   class class class wbr 4613  dom cdm 5074   DProd cdprd 18313 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-8 1989  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  ax-un 6902 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-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  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-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-oprab 6608  df-mpt2 6609  df-dprd 18315 This theorem is referenced by:  dprddomcld  18321  dprdsubg  18344
 Copyright terms: Public domain W3C validator