Theorem ssind 4063

Description: A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
ssind.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
ssind.2	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Assertion

Ref	Expression
ssind	⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶))

Proof of Theorem ssind

Step	Hyp	Ref	Expression
1		ssind.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		ssind.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
3	1, 2	jca 507	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶))
4		ssin 4061	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))
5	3, 4	sylib 210	1 ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 386   ∩ cin 3797   ⊆ wss 3798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-in 3805  df-ss 3812

This theorem is referenced by:  mreexexlem3d  16666  isacs1i  16677  rescabs  16852  funcres2c  16920  lsmmod  18446  gsumzres  18670  gsumzsubmcl  18678  gsum2d  18731  issubdrg  19168  lspdisj  19491  mplind  19869  ntrin  21243  elcls  21255  neitr  21362  restcls  21363  lmss  21480  xkoinjcn  21868  trfg  22072  trust  22410  utoptop  22415  restutop  22418  isngp2  22778  lebnumii  23142  causs  23473  dvreslem  24079  c1lip3  24168  ssjo  28857  dmdbr5  29718  mdslj2i  29730  mdsl2bi  29733  mdslmd1lem2  29736  mdsymlem5  29817  difininv  29898  bnj1286  31629  mclsind  32009  neiin  32860  topmeet  32892  fnemeet2  32895  bj-restpw  33567  bj-restb  33569  bj-restuni2  33573  idresssidinxp  34627  pmod1i  35922  dihmeetlem1N  37364  dihglblem5apreN  37365  dochdmj1  37464  mapdin  37736  baerlem3lem2  37784  baerlem5alem2  37785  baerlem5blem2  37786  trrelind  38797  isotone2  39186  nzin  39356  inmap  40206  islptre  40644  limccog  40645  limcresiooub  40667  limcresioolb  40668  limsupresxr  40791  liminfresxr  40792  liminfvalxr  40808  fourierdlem48  41163  fourierdlem49  41164  fourierdlem113  41228  pimiooltgt  41713  pimdecfgtioc  41717  pimincfltioc  41718  pimdecfgtioo  41719  pimincfltioo  41720  sssmf  41739  smflimlem2  41772  smfsuplem1  41809  setrec2fun  43348