Theorem hlrel 31522

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 31520	. . 3 ⊢ (𝑥 ∈ CHilOLD → 𝑥 ∈ CBan)
2	1	ssriv 3917	. 2 ⊢ CHilOLD ⊆ CBan
3		bnrel 31499	. 2 ⊢ Rel CBan
4		relss 5763	. 2 ⊢ (CHilOLD ⊆ CBan → (Rel CBan → Rel CHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CHilOLD

This proof depends on syntax axioms:   ⊆ wss 3900  Rel wrel 5660  CBanccbn 31494  CHil_OLDchlo 31517

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-iota 6512  df-fv 6566  df-oprab 7447  df-nv 31224  df-cbn 31495  df-hlo 31518

This theorem is used by: (None)