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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 7183 | . 2 ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) | |
2 | 1 | mpt2ndm0 6773 | 1 ⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∅c0 3874 {csn 4125 dom cdm 5038 “ cima 5041 (class class class)co 6549 supp csupp 7182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-supp 7183 |
This theorem is referenced by: suppssdm 7195 suppun 7202 extmptsuppeq 7206 funsssuppss 7208 fczsupp0 7211 suppss 7212 suppssov1 7214 suppss2 7216 suppssfv 7218 supp0cosupp0 7221 imacosupp 7222 fsuppun 8177 |
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