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Mirrors > Home > MPE Home > Th. List > supp0prc | Structured version Visualization version GIF version |
Description: The support of a class is empty if either the class or the "zero" is a proper class. (Contributed by AV, 28-May-2019.) |
Ref | Expression |
---|---|
supp0prc | ⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-supp 7882 | . 2 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) | |
2 | 1 | mpondm0 7424 | 1 ⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 {crab 3055 Vcvv 3398 ∅c0 4223 {csn 4527 dom cdm 5536 “ cima 5539 (class class class)co 7191 supp csupp 7881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-dm 5546 df-iota 6316 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-supp 7882 |
This theorem is referenced by: suppssdm 7897 suppun 7904 extmptsuppeq 7908 funsssuppss 7910 fczsupp0 7913 suppss 7914 suppssOLD 7915 suppssov1 7918 suppss2 7920 suppssfv 7922 suppco 7926 fsuppun 8982 |
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