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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 28923 | . . 3 ⊢ (𝑥 ∈ CHilOLD → 𝑥 ∈ CBan) | |
2 | 1 | ssriv 3891 | . 2 ⊢ CHilOLD ⊆ CBan |
3 | bnrel 28902 | . 2 ⊢ Rel CBan | |
4 | relss 5638 | . 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 3853 Rel wrel 5541 CBanccbn 28897 CHilOLDchlo 28920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-iota 6316 df-fv 6366 df-oprab 7195 df-nv 28627 df-cbn 28898 df-hlo 28921 |
This theorem is referenced by: (None) |
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