Theorem ssinss1d 39214

Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
ssinss1d.1	|- ( ph -> A C_ C )

Assertion

Ref	Expression
ssinss1d	|- ( ph -> ( A i^i B ) C_ C )

Proof of Theorem ssinss1d

Step	Hyp	Ref	Expression
1		ssinss1d.1	. 2 |- ( ph -> A C_ C )
2		ssinss1 3841	. 2 |- ( A C_ C -> ( A i^i B ) C_ C )
3	1, 2	syl 17	1 |- ( ph -> ( A i^i B ) C_ C )

Syntax hints:   -> wi 4   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:  ssinss2d  39228  ovolsplit  40205  caragenuncllem  40726  carageniuncllem1  40735  ovnsplit  40862  vonvolmbllem  40874  vonvolmbl  40875