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Mirrors > Home > MPE Home > Th. List > coemptyd | Structured version Visualization version GIF version |
Description: Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
coemptyd.1 | ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) |
Ref | Expression |
---|---|
coemptyd | ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coemptyd.1 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) | |
2 | coeq0 6078 | . 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅) | |
3 | 1, 2 | sylibr 237 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∩ cin 3853 ∅c0 4221 dom cdm 5517 ran crn 5518 ∘ ccom 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-br 5026 df-opab 5088 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 |
This theorem is referenced by: xptrrel 14372 cosnopne 30536 coeq0i 40052 conrel1d 40722 conrel2d 40723 clsneibex 41163 neicvgbex 41173 |
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