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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 7831 | . 2 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) | |
2 | 1 | mpondm0 7386 | 1 ⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 {crab 3142 Vcvv 3494 ∅c0 4291 {csn 4567 dom cdm 5555 “ cima 5558 (class class class)co 7156 supp csupp 7830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-dm 5565 df-iota 6314 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-supp 7831 |
This theorem is referenced by: suppssdm 7843 suppun 7850 extmptsuppeq 7854 funsssuppss 7856 fczsupp0 7859 suppss 7860 suppssov1 7862 suppss2 7864 suppssfv 7866 suppco 7870 supp0cosupp0OLD 7873 imacosuppOLD 7875 fsuppun 8852 |
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