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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 5307 | . . 3 ⊢ ∅ ∈ V | |
2 | suppval 8167 | . . 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 5923 | . . 3 ⊢ dom ∅ = ∅ | |
5 | rabeq 3443 | . . 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 4383 | . . 3 ⊢ {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅ | |
8 | 7 | a1i 11 | . 2 ⊢ (𝑍 ∈ 𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅) |
9 | 3, 6, 8 | 3eqtrd 2772 | 1 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 {crab 3429 Vcvv 3471 ∅c0 4323 {csn 4629 dom cdm 5678 “ cima 5681 (class class class)co 7420 supp csupp 8165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-supp 8166 |
This theorem is referenced by: 0fsupp 9413 gsumval3 19861 |
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