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Mirrors > Home > MPE Home > Th. List > supp0 | Structured version Visualization version GIF version |
Description: The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.) |
Ref | Expression |
---|---|
supp0 | ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5175 | . . 3 ⊢ ∅ ∈ V | |
2 | suppval 7815 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑍 ∈ 𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}}) | |
3 | 1, 2 | mpan 689 | . 2 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}}) |
4 | dm0 5754 | . . 3 ⊢ dom ∅ = ∅ | |
5 | rabeq 3431 | . . 3 ⊢ (dom ∅ = ∅ → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}}) | |
6 | 4, 5 | mp1i 13 | . 2 ⊢ (𝑍 ∈ 𝑊 → {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}}) |
7 | rab0 4291 | . . 3 ⊢ {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅ | |
8 | 7 | a1i 11 | . 2 ⊢ (𝑍 ∈ 𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅) |
9 | 3, 6, 8 | 3eqtrd 2837 | 1 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 Vcvv 3441 ∅c0 4243 {csn 4525 dom cdm 5519 “ cima 5522 (class class class)co 7135 supp csupp 7813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-supp 7814 |
This theorem is referenced by: 0fsupp 8839 gsumval3 19020 |
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