Theorem unissd 4866

Description: Subclass relationship for subclass union. Deduction form of uniss 4864. (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 4864	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3897  ∪ cuni 4856

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 3914  df-uni 4857

This theorem is referenced by:  unieq  4867  dffv2  6917  onfununi  8261  fiuni  9312  dfac2a  10021  incexc  15744  incexc2  15745  isacs1i  17563  isacs3lem  18448  acsmapd  18460  acsmap2d  18461  dprdres  19942  dprd2da  19956  eltg3i  22876  unitg  22882  tgss  22883  tgcmp  23316  cmpfi  23323  alexsubALTlem4  23965  ptcmplem3  23969  ustbas2  24140  uniioombllem3  25513  madess  27821  oldss  27823  shsupunss  31326  locfinref  33854  cmpcref  33863  dya2iocucvr  34297  omssubadd  34313  carsggect  34331  carsgclctun  34334  cvmscld  35317  fnemeet1  36410  fnejoin1  36412  onsucsuccmpi  36487  heibor1  37849  heiborlem10  37859  hbt  43222  pwsal  46412  prsal  46415  intsaluni  46426  caragenuni  46608  caragendifcl  46611  cnfsmf  46837  smfsssmf  46840  smfpimbor1lem2  46896  toplatglb  49100  setrecsss  49801