Theorem hlrel 30872

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 CHilOLD

Proof of Theorem hlrel

Step	Hyp	Ref	Expression
1		hlobn 30870	. . 3 ⊢ (𝑥 ∈ CHilOLD → 𝑥 ∈ CBan)
2	1	ssriv 3934	. 2 ⊢ CHilOLD ⊆ CBan
3		bnrel 30849	. 2 ⊢ Rel CBan
4		relss 5726	. 2 ⊢ (CHilOLD ⊆ CBan → (Rel CBan → Rel CHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CHilOLD

Syntax hints:   ⊆ wss 3898  Rel wrel 5624  CBanccbn 30844  CHil_OLDchlo 30867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-iota 6442  df-fv 6494  df-oprab 7356  df-nv 30574  df-cbn 30845  df-hlo 30868

This theorem is referenced by: (None)