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Theorem hlrel 21415
Description: The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hlrel  |-  Rel  CHil OLD

Proof of Theorem hlrel
StepHypRef Expression
1 hlobn 21413 . . 3  |-  ( x  e.  CHil OLD  ->  x  e. 
CBan )
21ssriv 3145 . 2  |-  CHil OLD  C_ 
CBan
3 bnrel 21392 . 2  |-  Rel  CBan
4 relss 4749 . 2  |-  ( CHil
OLD  C_  CBan  ->  ( Rel 
CBan  ->  Rel  CHil OLD )
)
52, 3, 4mp2 19 1  |-  Rel  CHil OLD
Colors of variables: wff set class
Syntax hints:    C_ wss 3113   Rel wrel 4652   CBanccbn 21387   CHil OLDchlo 21410
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-xp 4661  df-rel 4662  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fv 4675  df-oprab 5782  df-nv 21094  df-cbn 21388  df-hlo 21411
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