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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 28649 | . . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec) | |
2 | 1 | ssriv 3919 | . 2 ⊢ CBan ⊆ NrmCVec |
3 | nvrel 28385 | . 2 ⊢ Rel NrmCVec | |
4 | relss 5620 | . 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 3881 Rel wrel 5524 NrmCVeccnv 28367 CBanccbn 28645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-iota 6283 df-fv 6332 df-oprab 7139 df-nv 28375 df-cbn 28646 |
This theorem is referenced by: hlrel 28673 |
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