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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 5203 | . . 3 ⊢ ∅ ∈ V | |
2 | suppval 7826 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑍 ∈ 𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}}) | |
3 | 1, 2 | mpan 688 | . 2 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}}) |
4 | dm0 5784 | . . 3 ⊢ dom ∅ = ∅ | |
5 | rabeq 3483 | . . 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 4336 | . . 3 ⊢ {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅ | |
8 | 7 | a1i 11 | . 2 ⊢ (𝑍 ∈ 𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅) |
9 | 3, 6, 8 | 3eqtrd 2860 | 1 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 Vcvv 3494 ∅c0 4290 {csn 4560 dom cdm 5549 “ cima 5552 (class class class)co 7150 supp csupp 7824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-supp 7825 |
This theorem is referenced by: 0fsupp 8849 gsumval3 19021 |
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