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Theorem dprddomprc 20000
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 3037 . . 3 (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
2 dmexg 7914 . . . 4 (𝑆 ∈ V → dom 𝑆 ∈ V)
32con3i 154 . . 3 (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V)
41, 3sylbi 216 . 2 (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V)
5 reldmdprd 19997 . . 3 Rel dom DProd
65brrelex2i 5739 . 2 (𝐺dom DProd 𝑆𝑆 ∈ V)
74, 6nsyl 140 1 (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2099  wnel 3036  Vcvv 3462   class class class wbr 5153  dom cdm 5682   DProd cdprd 19993
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-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-oprab 7428  df-mpo 7429  df-dprd 19995
This theorem is referenced by:  dprddomcld  20001  dprdsubg  20024
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