Theorem hlrel 31379

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 31377	. . 3 ⊢ (𝑥 ∈ CHilOLD → 𝑥 ∈ CBan)
2	1	ssriv 3938	. 2 ⊢ CHilOLD ⊆ CBan
3		bnrel 31356	. 2 ⊢ Rel CBan
4		relss 5766	. 2 ⊢ (CHilOLD ⊆ CBan → (Rel CBan → Rel CHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CHilOLD

This proof depends on syntax axioms:   ⊆ wss 3902  Rel wrel 5664  CBanccbn 31351  CHil_OLDchlo 31374

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-iota 6493  df-fv 6545  df-oprab 7421  df-nv 31081  df-cbn 31352  df-hlo 31375

This theorem is used by: (None)