Theorem ineqcom 34005

Description: Two ways of saying that two classes are disjoint (when C = (/): ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )). (Contributed by Peter Mazsa, 22-Mar-2017.)

Assertion

Ref	Expression
ineqcom	|- ( ( A i^i B ) = C <-> ( B i^i A ) = C )

Proof of Theorem ineqcom

Step	Hyp	Ref	Expression
1		incom 3805	. 2 |- ( A i^i B ) = ( B i^i A )
2	1	eqeq1i 2627	1 |- ( ( A i^i B ) = C <-> ( B i^i A ) = C )

Syntax hints:   <-> wb 196   = wceq 1483   i^i cin 3573

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

This theorem is referenced by:  ineqcomi  34006