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Mirrors > Home > MPE Home > Th. List > hlrel | Structured version Visualization version GIF version |
Description: The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlrel | ⊢ Rel CHilOLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlobn 29244 | . . 3 ⊢ (𝑥 ∈ CHilOLD → 𝑥 ∈ CBan) | |
2 | 1 | ssriv 3930 | . 2 ⊢ CHilOLD ⊆ CBan |
3 | bnrel 29223 | . 2 ⊢ Rel CBan | |
4 | relss 5691 | . 2 ⊢ (CHilOLD ⊆ CBan → (Rel CBan → Rel CHilOLD)) | |
5 | 2, 3, 4 | mp2 9 | 1 ⊢ Rel CHilOLD |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3892 Rel wrel 5594 CBanccbn 29218 CHilOLDchlo 29241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-xp 5595 df-rel 5596 df-iota 6389 df-fv 6439 df-oprab 7273 df-nv 28948 df-cbn 29219 df-hlo 29242 |
This theorem is referenced by: (None) |
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