Theorem ackbij1lem1 9042

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

Assertion

Ref	Expression
ackbij1lem1	|- ( -. A e. B -> ( B i^i suc A ) = ( B i^i A ) )

Proof of Theorem ackbij1lem1

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	2, 3	eqtri 2644	. 2 |- ( B i^i suc A ) = ( ( B i^i A ) u. ( B i^i { A } ) )
5		disjsn 4246	. . . . 5 |- ( ( B i^i { A } ) = (/) <-> -. A e. B )
6	5	biimpri 218	. . . 4 |- ( -. A e. B -> ( B i^i { A } ) = (/) )
7	6	uneq2d 3767	. . 3 |- ( -. A e. B -> ( ( B i^i A ) u. ( B i^i { A } ) ) = ( ( B i^i A ) u. (/) ) )
8		un0 3967	. . 3 |- ( ( B i^i A ) u. (/) ) = ( B i^i A )
9	7, 8	syl6eq 2672	. 2 |- ( -. A e. B -> ( ( B i^i A ) u. ( B i^i { A } ) ) = ( B i^i A ) )
10	4, 9	syl5eq 2668	1 |- ( -. A e. B -> ( B i^i suc A ) = ( B i^i A ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   (/)c0 3915   {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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-suc 5729

This theorem is referenced by:  ackbij1lem15  9056  ackbij1lem16  9057