Theorem ssini 4261

Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)

Hypotheses

Ref	Expression
ssini.1	⊢ 𝐴 ⊆ 𝐵
ssini.2	⊢ 𝐴 ⊆ 𝐶

Assertion

Ref	Expression
ssini	⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶)

Proof of Theorem ssini

Step	Hyp	Ref	Expression
1		ssini.1	. . 3 ⊢ 𝐴 ⊆ 𝐵
2		ssini.2	. . 3 ⊢ 𝐴 ⊆ 𝐶
3	1, 2	pm3.2i 470	. 2 ⊢ (𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶)
4		ssin 4260	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))
5	3, 4	mpbi 230	1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶)

Syntax hints:   ∧ wa 395   ∩ cin 3975   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-in 3983  df-ss 3993

This theorem is referenced by:  inv1  4421  cnvrescnv  6226  hartogslem1  9611  xptrrel  15029  fbasrn  23913  limciun  25949  hlimcaui  31268  chdmm1i  31509  chm0i  31522  ledii  31568  lejdii  31570  mayetes3i  31761  mdslj2i  32352  mdslmd2i  32362  sumdmdlem2  32451  sigapildsys  34126  ssoninhaus  36414  bj-disj2r  36994  bj-idres  37126  bj-rvecsscvec  37270  icomnfinre  45470  fouriersw  46152  sge0split  46330