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Theorem chle0i 27695
 Description: No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
Hypothesis
Ref Expression
ch0le.1 𝐴C
Assertion
Ref Expression
chle0i (𝐴 ⊆ 0𝐴 = 0)

Proof of Theorem chle0i
StepHypRef Expression
1 ch0le.1 . 2 𝐴C
2 chle0 27686 . 2 (𝐴C → (𝐴 ⊆ 0𝐴 = 0))
31, 2ax-mp 5 1 (𝐴 ⊆ 0𝐴 = 0)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   Cℋ cch 27170  0ℋc0h 27176 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-sh 27448  df-ch 27462  df-ch0 27494 This theorem is referenced by:  chj00i  27730  chsup0  27791  spansnm0i  27893  largei  28510
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