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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 5313 | . . 3 ⊢ ∅ ∈ V | |
2 | suppval 8186 | . . 3 ⊢ ((∅ ∈ V ∧ 𝑍 ∈ 𝑊) → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}}) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = {𝑖 ∈ dom ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}}) |
4 | dm0 5934 | . . 3 ⊢ dom ∅ = ∅ | |
5 | rabeq 3448 | . . 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 4392 | . . 3 ⊢ {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅ | |
8 | 7 | a1i 11 | . 2 ⊢ (𝑍 ∈ 𝑊 → {𝑖 ∈ ∅ ∣ (∅ “ {𝑖}) ≠ {𝑍}} = ∅) |
9 | 3, 6, 8 | 3eqtrd 2779 | 1 ⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {crab 3433 Vcvv 3478 ∅c0 4339 {csn 4631 dom cdm 5689 “ cima 5692 (class class class)co 7431 supp csupp 8184 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-supp 8185 |
This theorem is referenced by: 0fsupp 9428 gsumval3 19940 |
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