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Mirrors > Home > MPE Home > Th. List > bnrel | Structured version Visualization version GIF version |
Description: The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnrel | ⊢ Rel CBan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnnv 28294 | . . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec) | |
2 | 1 | ssriv 3825 | . 2 ⊢ CBan ⊆ NrmCVec |
3 | nvrel 28029 | . 2 ⊢ Rel NrmCVec | |
4 | relss 5454 | . 2 ⊢ (CBan ⊆ NrmCVec → (Rel NrmCVec → Rel CBan)) | |
5 | 2, 3, 4 | mp2 9 | 1 ⊢ Rel CBan |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3792 Rel wrel 5360 NrmCVeccnv 28011 CBanccbn 28290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-rel 5362 df-iota 6099 df-fv 6143 df-oprab 6926 df-nv 28019 df-cbn 28291 |
This theorem is referenced by: hlrel 28318 |
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