Theorem ackbij1lem2 9043

Description: Lemma for ackbij2 9065. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Assertion

Ref	Expression
ackbij1lem2	|- ( A e. B -> ( B i^i suc A ) = ( { A } u. ( B i^i A ) ) )

Proof of Theorem ackbij1lem2

Step	Hyp	Ref	Expression
1		df-suc 5729	. . . 4 |- suc A = ( A u. { A } )
2	1	ineq2i 3811	. . 3 |- ( B i^i suc A ) = ( B i^i ( A u. { A } ) )
3		indi 3873	. . 3 |- ( B i^i ( A u. { A } ) ) = ( ( B i^i A ) u. ( B i^i { A } ) )
4		uncom 3757	. . 3 |- ( ( B i^i A ) u. ( B i^i { A } ) ) = ( ( B i^i { A } ) u. ( B i^i A ) )
5	2, 3, 4	3eqtri 2648	. 2 |- ( B i^i suc A ) = ( ( B i^i { A } ) u. ( B i^i A ) )
6		snssi 4339	. . . 4 |- ( A e. B -> { A } C_ B )
7		sseqin2 3817	. . . 4 |- ( { A } C_ B <-> ( B i^i { A } ) = { A } )
8	6, 7	sylib 208	. . 3 |- ( A e. B -> ( B i^i { A } ) = { A } )
9	8	uneq1d 3766	. 2 |- ( A e. B -> ( ( B i^i { A } ) u. ( B i^i A ) ) = ( { A } u. ( B i^i A ) ) )
10	5, 9	syl5eq 2668	1 |- ( A e. B -> ( B i^i suc A ) = ( { A } u. ( B i^i A ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   suc csuc 5725

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-3an 1039  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-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-suc 5729

This theorem is referenced by:  ackbij1lem15  9056  ackbij1lem16  9057