Theorem disjdifr 3584

Description: A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.)

Assertion

Ref	Expression
disjdifr	|- ( ( B \ A ) i^i A ) = (/)

Proof of Theorem disjdifr

Step	Hyp	Ref	Expression
1		disjdif 3583	. 2 |- ( A i^i ( B \ A ) ) = (/)
2	1	ineqcomi 3415	1 |- ( ( B \ A ) i^i A ) = (/)

Syntax hints:   = wceq 1398   \ cdif 3210   i^i cin 3212   (/)c0 3510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3215  df-in 3219  df-ss 3226  df-nul 3511

This theorem is referenced by:  hashfibclem  11210