Theorem ssini 4139

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 4138	. 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   ⊆ wss 3900   ∩ cin 4092

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-in 4094

This theorem is used by:  inv1  4351  uniin  4894  rnin  6149  cnvrescnv  6200  hartogslem1  9563  xptrrel  15162  fbasrn  24233  limciun  26244  hlimcaui  31868  chdmm1i  32109  chm0i  32122  ledii  32168  lejdii  32170  mayetes3i  32361  mdslj2i  32952  mdslmd2i  32962  sumdmdlem2  33051  sigapildsys  34865  ssoninhaus  37414  bj-disj2r  38119  bj-idres  38273  bj-rvecsscvec  38417  icomnfinre  46769  fouriersw  47446  sge0split  47624  numtowerdt  48121