Theorem inss 3842

Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)

Assertion

Ref	Expression
inss	|- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C )

Proof of Theorem inss

Step	Hyp	Ref	Expression
1		ssinss1 3841	. 2 |- ( A C_ C -> ( A i^i B ) C_ C )
2		incom 3805	. . 3 |- ( A i^i B ) = ( B i^i A )
3		ssinss1 3841	. . 3 |- ( B C_ C -> ( B i^i A ) C_ C )
4	2, 3	syl5eqss 3649	. 2 |- ( B C_ C -> ( A i^i B ) C_ C )
5	1, 4	jaoi 394	1 |- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C )

Syntax hints:   -> wi 4   \/ wo 383   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:  pmatcoe1fsupp  20506  ppttop  20811  inindif  29353  iunrelexp0  37994  ntrclsk3  38368  icccncfext  40100