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Theorem dprdval0prc 19394
Description: The internal direct product of a family of subgroups indexed by a proper class is empty. (Contributed by AV, 13-Jul-2019.)
Assertion
Ref Expression
dprdval0prc (dom 𝑆 ∉ V → (𝐺 DProd 𝑆) = ∅)

Proof of Theorem dprdval0prc
StepHypRef Expression
1 df-nel 3047 . . 3 (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
2 dmexg 7686 . . . 4 (𝑆 ∈ V → dom 𝑆 ∈ V)
32con3i 157 . . 3 (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V)
41, 3sylbi 220 . 2 (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V)
5 reldmdprd 19389 . . 3 Rel dom DProd
65ovprc2 7258 . 2 𝑆 ∈ V → (𝐺 DProd 𝑆) = ∅)
74, 6syl 17 1 (dom 𝑆 ∉ V → (𝐺 DProd 𝑆) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2110  wnel 3046  Vcvv 3413  c0 4242  dom cdm 5556  (class class class)co 7218   DProd cdprd 19385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-br 5059  df-opab 5121  df-xp 5562  df-rel 5563  df-cnv 5564  df-dm 5566  df-rn 5567  df-iota 6343  df-fv 6393  df-ov 7221  df-oprab 7222  df-mpo 7223  df-dprd 19387
This theorem is referenced by: (None)
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