Theorem hlrel 31220

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 31218	. . 3 ⊢ (𝑥 ∈ CHilOLD → 𝑥 ∈ CBan)
2	1	ssriv 3942	. 2 ⊢ CHilOLD ⊆ CBan
3		bnrel 31197	. 2 ⊢ Rel CBan
4		relss 5770	. 2 ⊢ (CHilOLD ⊆ CBan → (Rel CBan → Rel CHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CHilOLD

Syntax hints:   ⊆ wss 3906  Rel wrel 5668  CBanccbn 31192  CHil_OLDchlo 31215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-iota 6494  df-fv 6546  df-oprab 7416  df-nv 30922  df-cbn 31193  df-hlo 31216

This theorem is referenced by: (None)