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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 2783 | . . 3 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
2 | dmexg 6989 | . . . 4 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V) | |
3 | 2 | con3i 149 | . . 3 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V) |
4 | 1, 3 | sylbi 206 | . 2 ⊢ (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V) |
5 | reldmdprd 18219 | . . 3 ⊢ Rel dom DProd | |
6 | 5 | ovprc2 6583 | . 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 1475 ∈ wcel 1977 ∉ wnel 2781 Vcvv 3173 ∅c0 3874 dom cdm 5038 (class class class)co 6549 DProd cdprd 18215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-dprd 18217 |
This theorem is referenced by: (None) |
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