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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 28637 | . . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec) | |
2 | 1 | ssriv 3970 | . 2 ⊢ CBan ⊆ NrmCVec |
3 | nvrel 28373 | . 2 ⊢ Rel NrmCVec | |
4 | relss 5650 | . 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 3935 Rel wrel 5554 NrmCVeccnv 28355 CBanccbn 28633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-iota 6308 df-fv 6357 df-oprab 7154 df-nv 28363 df-cbn 28634 |
This theorem is referenced by: hlrel 28661 |
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