Theorem hlrel 30819

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 30817	. . 3 ⊢ (𝑥 ∈ CHilOLD → 𝑥 ∈ CBan)
2	1	ssriv 3950	. 2 ⊢ CHilOLD ⊆ CBan
3		bnrel 30796	. 2 ⊢ Rel CBan
4		relss 5744	. 2 ⊢ (CHilOLD ⊆ CBan → (Rel CBan → Rel CHilOLD))
5	2, 3, 4	mp2 9	1 ⊢ Rel CHilOLD

Syntax hints:   ⊆ wss 3914  Rel wrel 5643  CBanccbn 30791  CHil_OLDchlo 30814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-iota 6464  df-fv 6519  df-oprab 7391  df-nv 30521  df-cbn 30792  df-hlo 30815

This theorem is referenced by: (None)