Theorem inindif 29353

Description: See inundif 4046. (Contributed by Thierry Arnoux, 13-Sep-2017.)

Assertion

Ref	Expression
inindif	|- ( ( A i^i C ) i^i ( A \ C ) ) = (/)

Proof of Theorem inindif

Step	Hyp	Ref	Expression
1		inss2 3834	. . . 4 |- ( A i^i C ) C_ C
2	1	orci 405	. . 3 |- ( ( A i^i C ) C_ C \/ A C_ C )
3		inss 3842	. . 3 |- ( ( ( A i^i C ) C_ C \/ A C_ C ) -> ( ( A i^i C ) i^i A ) C_ C )
4	2, 3	ax-mp 5	. 2 |- ( ( A i^i C ) i^i A ) C_ C
5		inssdif0 3947	. 2 |- ( ( ( A i^i C ) i^i A ) C_ C <-> ( ( A i^i C ) i^i ( A \ C ) ) = (/) )
6	4, 5	mpbi 220	1 |- ( ( A i^i C ) i^i ( A \ C ) ) = (/)

Syntax hints:   \/ wo 383   = wceq 1483   \ cdif 3571   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:  resf1o  29505  gsummptres  29784  indsumin  30084  measunl  30279  carsgclctun  30383  probdif  30482  hgt750lemd  30726