Theorem ssini 4251

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 4250	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))
5	3, 4	mpbi 230	1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶)

Syntax hints:   ∧ wa 395   ∩ cin 3965   ⊆ wss 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3483  df-in 3973  df-ss 3983

This theorem is referenced by:  inv1  4407  cnvrescnv  6223  hartogslem1  9589  xptrrel  15025  fbasrn  23917  limciun  25955  hlimcaui  31281  chdmm1i  31522  chm0i  31535  ledii  31581  lejdii  31583  mayetes3i  31774  mdslj2i  32365  mdslmd2i  32375  sumdmdlem2  32464  sigapildsys  34157  ssoninhaus  36443  bj-disj2r  37023  bj-idres  37155  bj-rvecsscvec  37299  icomnfinre  45534  fouriersw  46215  sge0split  46393