Theorem ordintdif 5774

Description: If B is smaller than A, then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)

Assertion

Ref	Expression
ordintdif	|- ( ( Ord A /\ Ord B /\ ( A \ B ) =/= (/) ) -> B = |^| ( A \ B ) )

Proof of Theorem ordintdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssdif0 3942	. . 3 |- ( A C_ B <-> ( A \ B ) = (/) )
2	1	necon3bbii 2841	. 2 |- ( -. A C_ B <-> ( A \ B ) =/= (/) )
3		dfdif2 3583	. . . 4 |- ( A \ B ) = { x e. A | -. x e. B }
4	3	inteqi 4479	. . 3 |- |^| ( A \ B ) = |^| { x e. A | -. x e. B }
5		ordtri1 5756	. . . . . 6 |- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> -. B e. A ) )
6	5	con2bid 344	. . . . 5 |- ( ( Ord A /\ Ord B ) -> ( B e. A <-> -. A C_ B ) )
7		id 22	. . . . . . . . . . 11 |- ( Ord B -> Ord B )
8		ordelord 5745	. . . . . . . . . . 11 |- ( ( Ord A /\ x e. A ) -> Ord x )
9		ordtri1 5756	. . . . . . . . . . 11 |- ( ( Ord B /\ Ord x ) -> ( B C_ x <-> -. x e. B ) )
10	7, 8, 9	syl2anr 495	. . . . . . . . . 10 |- ( ( ( Ord A /\ x e. A ) /\ Ord B ) -> ( B C_ x <-> -. x e. B ) )
11	10	an32s 846	. . . . . . . . 9 |- ( ( ( Ord A /\ Ord B ) /\ x e. A ) -> ( B C_ x <-> -. x e. B ) )
12	11	rabbidva 3188	. . . . . . . 8 |- ( ( Ord A /\ Ord B ) -> { x e. A | B C_ x } = { x e. A | -. x e. B } )
13	12	inteqd 4480	. . . . . . 7 |- ( ( Ord A /\ Ord B ) -> |^| { x e. A | B C_ x } = |^| { x e. A | -. x e. B } )
14		intmin 4497	. . . . . . 7 |- ( B e. A -> |^| { x e. A | B C_ x } = B )
15	13, 14	sylan9req 2677	. . . . . 6 |- ( ( ( Ord A /\ Ord B ) /\ B e. A ) -> |^| { x e. A | -. x e. B } = B )
16	15	ex 450	. . . . 5 |- ( ( Ord A /\ Ord B ) -> ( B e. A -> |^| { x e. A | -. x e. B } = B ) )
17	6, 16	sylbird 250	. . . 4 |- ( ( Ord A /\ Ord B ) -> ( -. A C_ B -> |^| { x e. A | -. x e. B } = B ) )
18	17	3impia 1261	. . 3 |- ( ( Ord A /\ Ord B /\ -. A C_ B ) -> |^| { x e. A | -. x e. B } = B )
19	4, 18	syl5req 2669	. 2 |- ( ( Ord A /\ Ord B /\ -. A C_ B ) -> B = |^| ( A \ B ) )
20	2, 19	syl3an3br 1367	1 |- ( ( Ord A /\ Ord B /\ ( A \ B ) =/= (/) ) -> B = |^| ( A \ B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   |^|cint 4475   Ord word 5722

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by: (None)