Theorem ssini 4120

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 474	. 2 ⊢ (𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶)
4		ssin 4119	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))
5	3, 4	mpbi 233	1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶)

Syntax hints:   ∧ wa 399   ∩ cin 3840   ⊆ wss 3841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399  df-in 3848  df-ss 3858

This theorem is referenced by:  inv1  4280  cnvrescnv  6021  hartogslem1  9072  xptrrel  14422  fbasrn  22628  limciun  24638  hlimcaui  29163  chdmm1i  29404  chm0i  29417  ledii  29463  lejdii  29465  mayetes3i  29656  mdslj2i  30247  mdslmd2i  30257  sumdmdlem2  30346  sigapildsys  31692  ssoninhaus  34267  bj-disj2r  34830  bj-idres  34941  bj-rvecsscvec  35084  icomnfinre  42614  fouriersw  43298  sge0split  43473