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Mirrors > Home > MPE Home > Th. List > dprdval0prc | Structured version Visualization version GIF version |
Description: The internal direct product of a family of subgroups indexed by a proper class is empty. (Contributed by AV, 13-Jul-2019.) |
Ref | Expression |
---|---|
dprdval0prc | ⊢ (dom 𝑆 ∉ V → (𝐺 DProd 𝑆) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3047 | . . 3 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
2 | dmexg 7686 | . . . 4 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V) | |
3 | 2 | con3i 157 | . . 3 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V) |
4 | 1, 3 | sylbi 220 | . 2 ⊢ (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V) |
5 | reldmdprd 19389 | . . 3 ⊢ Rel dom DProd | |
6 | 5 | ovprc2 7258 | . 2 ⊢ (¬ 𝑆 ∈ V → (𝐺 DProd 𝑆) = ∅) |
7 | 4, 6 | syl 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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