Theorem ssdisj 4026

Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

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

Proof of Theorem ssdisj

Step	Hyp	Ref	Expression
1		ssrin 3838	. . 3 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
2		eqimss 3657	. . 3 |- ( ( B i^i C ) = (/) -> ( B i^i C ) C_ (/) )
3	1, 2	sylan9ss 3616	. 2 |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) C_ (/) )
4		ss0 3974	. 2 |- ( ( A i^i C ) C_ (/) -> ( A i^i C ) = (/) )
5	3, 4	syl 17	1 |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   i^i cin 3573   C_ wss 3574   (/)c0 3915

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-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  djudisj  5561  fimacnvdisj  6083  marypha1lem  8339  ackbij1lem16  9057  ackbij1lem18  9059  fin23lem20  9159  fin23lem30  9164  elcls3  20887  neindisj  20921  imadifxp  29414  ldgenpisyslem1  30226  chtvalz  30707  diophren  37377