Theorem ssini 4180

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 4179	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))
5	3, 4	mpbi 230	1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶)

Syntax hints:   ∧ wa 395   ∩ cin 3888   ⊆ wss 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-in 3896  df-ss 3906

This theorem is referenced by:  inv1  4338  cnvrescnv  6159  hartogslem1  9457  xptrrel  14942  fbasrn  23849  limciun  25861  hlimcaui  31307  chdmm1i  31548  chm0i  31561  ledii  31607  lejdii  31609  mayetes3i  31800  mdslj2i  32391  mdslmd2i  32401  sumdmdlem2  32490  sigapildsys  34306  ssoninhaus  36630  bj-disj2r  37335  bj-idres  37474  bj-rvecsscvec  37618  icomnfinre  45982  fouriersw  46659  sge0split  46837  nthrucw  47316