Theorem undifexmid 4117

Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3443 and undifdcss 6811 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)

Hypothesis

Ref	Expression
undifexmid.1	|- ( x C_ y <-> ( x u. ( y \ x ) ) = y )

Assertion

Ref	Expression
undifexmid	|- ( ph \/ -. ph )

Distinct variable group:   ph, x, y

Proof of Theorem undifexmid

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4055	. . . . 5 |- (/) e. _V
2	1	snid 3556	. . . 4 |- (/) e. { (/) }
3		ssrab2 3182	. . . . 5 |- { z e. { (/) } | ph } C_ { (/) }
4		p0ex 4112	. . . . . . 7 |- { (/) } e. _V
5	4	rabex 4072	. . . . . 6 |- { z e. { (/) } | ph } e. _V
6		sseq12 3122	. . . . . . 7 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( x C_ y <-> { z e. { (/) } | ph } C_ { (/) } ) )
7		simpl 108	. . . . . . . . 9 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> x = { z e. { (/) } | ph } )
8		simpr 109	. . . . . . . . . 10 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> y = { (/) } )
9	8, 7	difeq12d 3195	. . . . . . . . 9 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( y \ x ) = ( { (/) } \ { z e. { (/) } | ph } ) )
10	7, 9	uneq12d 3231	. . . . . . . 8 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( x u. ( y \ x ) ) = ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) )
11	10, 8	eqeq12d 2154	. . . . . . 7 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( ( x u. ( y \ x ) ) = y <-> ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) = { (/) } ) )
12	6, 11	bibi12d 234	. . . . . 6 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( ( x C_ y <-> ( x u. ( y \ x ) ) = y ) <-> ( { z e. { (/) } | ph } C_ { (/) } <-> ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) = { (/) } ) ) )
13		undifexmid.1	. . . . . 6 |- ( x C_ y <-> ( x u. ( y \ x ) ) = y )
14	5, 4, 12, 13	vtocl2 2741	. . . . 5 |- ( { z e. { (/) } | ph } C_ { (/) } <-> ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) = { (/) } )
15	3, 14	mpbi 144	. . . 4 |- ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) = { (/) }
16	2, 15	eleqtrri 2215	. . 3 |- (/) e. ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) )
17		elun 3217	. . 3 |- ( (/) e. ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) <-> ( (/) e. { z e. { (/) } | ph } \/ (/) e. ( { (/) } \ { z e. { (/) } | ph } ) ) )
18	16, 17	mpbi 144	. 2 |- ( (/) e. { z e. { (/) } | ph } \/ (/) e. ( { (/) } \ { z e. { (/) } | ph } ) )
19		biidd 171	. . . . . 6 |- ( z = (/) -> ( ph <-> ph ) )
20	19	elrab3 2841	. . . . 5 |- ( (/) e. { (/) } -> ( (/) e. { z e. { (/) } | ph } <-> ph ) )
21	2, 20	ax-mp 5	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
22	21	biimpi 119	. . 3 |- ( (/) e. { z e. { (/) } | ph } -> ph )
23		eldifn 3199	. . . 4 |- ( (/) e. ( { (/) } \ { z e. { (/) } | ph } ) -> -. (/) e. { z e. { (/) } | ph } )
24	23, 21	sylnib 665	. . 3 |- ( (/) e. ( { (/) } \ { z e. { (/) } | ph } ) -> -. ph )
25	22, 24	orim12i 748	. 2 |- ( ( (/) e. { z e. { (/) } | ph } \/ (/) e. ( { (/) } \ { z e. { (/) } | ph } ) ) -> ( ph \/ -. ph ) )
26	18, 25	ax-mp 5	1 |- ( ph \/ -. ph )

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   {crab 2420   \ cdif 3068   u. cun 3069   C_ wss 3071   (/)c0 3363   {csn 3527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533

This theorem is referenced by: (None)