Theorem ssini 3836

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 C_ B
ssini.2	|- A C_ C

Assertion

Ref	Expression
ssini	|- A C_ ( B i^i C )

Proof of Theorem ssini

Step	Hyp	Ref	Expression
1		ssini.1	. . 3 |- A C_ B
2		ssini.2	. . 3 |- A C_ C
3	1, 2	pm3.2i 471	. 2 |- ( A C_ B /\ A C_ C )
4		ssin 3835	. 2 |- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )
5	3, 4	mpbi 220	1 |- A C_ ( B i^i C )

Syntax hints:   /\ wa 384   i^i cin 3573   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  inv1  3970  hartogslem1  8447  xptrrel  13719  fbasrn  21688  limciun  23658  hlimcaui  28093  chdmm1i  28336  chm0i  28349  ledii  28395  lejdii  28397  mayetes3i  28588  mdslj2i  29179  mdslmd2i  29189  sumdmdlem2  29278  sigapildsys  30225  ssoninhaus  32447  bj-disj2r  33013  idinxpres  34088  icomnfinre  39779  fouriersw  40448  sge0split  40626