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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 28592 | . . 3 ⊢ (𝑥 ∈ CHilOLD → 𝑥 ∈ CBan) | |
2 | 1 | ssriv 3968 | . 2 ⊢ CHilOLD ⊆ CBan |
3 | bnrel 28571 | . 2 ⊢ Rel CBan | |
4 | relss 5649 | . 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 3933 Rel wrel 5553 CBanccbn 28566 CHilOLDchlo 28589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-iota 6307 df-fv 6356 df-oprab 7149 df-nv 28296 df-cbn 28567 df-hlo 28590 |
This theorem is referenced by: (None) |
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