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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 2898 | . . 3 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
2 | dmexg 7097 | . . . 4 ⊢ (𝑆 ∈ V → dom 𝑆 ∈ V) | |
3 | 2 | con3i 150 | . . 3 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝑆 ∈ V) |
4 | 1, 3 | sylbi 207 | . 2 ⊢ (dom 𝑆 ∉ V → ¬ 𝑆 ∈ V) |
5 | reldmdprd 18396 | . . 3 ⊢ Rel dom DProd | |
6 | 5 | ovprc2 6685 | . 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 1483 ∈ wcel 1990 ∉ wnel 2897 Vcvv 3200 ∅c0 3915 dom cdm 5114 (class class class)co 6650 DProd cdprd 18392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-dprd 18394 |
This theorem is referenced by: (None) |
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