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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 30794 | . . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec) | |
2 | 1 | ssriv 3983 | . 2 ⊢ CBan ⊆ NrmCVec |
3 | nvrel 30530 | . 2 ⊢ Rel NrmCVec | |
4 | relss 5778 | . 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 3947 Rel wrel 5678 NrmCVeccnv 30512 CBanccbn 30790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-xp 5679 df-rel 5680 df-iota 6496 df-fv 6552 df-oprab 7418 df-nv 30520 df-cbn 30791 |
This theorem is referenced by: hlrel 30818 |
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