Theorem ssini 4188

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 476	. 2 ⊢ (𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶)
4		ssin 4187	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))
5	3, 4	mpbi 233	1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶)

This proof depends on syntax axioms:   ∧ wa 401   ∩ cin 3901   ⊆ wss 3902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-ss 3919

This theorem is used by:  inv1  4351  uniin  4894  rnin  6141  cnvrescnv  6193  hartogslem1  9518  xptrrel  15057  fbasrn  24116  limciun  26128  hlimcaui  31725  chdmm1i  31966  chm0i  31979  ledii  32025  lejdii  32027  mayetes3i  32218  mdslj2i  32809  mdslmd2i  32819  sumdmdlem2  32908  sigapildsys  34681  ssoninhaus  37075  bj-disj2r  37780  bj-idres  37920  bj-rvecsscvec  38064  icomnfinre  46390  fouriersw  47067  sge0split  47245  numtowerdt  47742