Theorem unissd 4869

Description: Subclass relationship for subclass union. Deduction form of uniss 4867. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
unissd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
unissd	⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵)

Proof of Theorem unissd

Step	Hyp	Ref	Expression
1		unissd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		uniss 4867	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3902  ∪ cuni 4859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3919  df-uni 4860

This theorem is referenced by:  unieq  4870  dffv2  6917  onfununi  8261  fiuni  9312  dfac2a  10021  incexc  15744  incexc2  15745  isacs1i  17563  isacs3lem  18448  acsmapd  18460  acsmap2d  18461  dprdres  19943  dprd2da  19957  eltg3i  22877  unitg  22883  tgss  22884  tgcmp  23317  cmpfi  23324  alexsubALTlem4  23966  ptcmplem3  23970  ustbas2  24141  uniioombllem3  25514  madess  27822  oldss  27824  shsupunss  31324  locfinref  33852  cmpcref  33861  dya2iocucvr  34295  omssubadd  34311  carsggect  34329  carsgclctun  34332  cvmscld  35315  fnemeet1  36406  fnejoin1  36408  onsucsuccmpi  36483  heibor1  37856  heiborlem10  37866  hbt  43169  pwsal  46359  prsal  46362  intsaluni  46373  caragenuni  46555  caragendifcl  46558  cnfsmf  46784  smfsssmf  46787  smfpimbor1lem2  46843  toplatglb  49038  setrecsss  49739