Theorem undifexmid 4208

Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3518 and undifdcss 6941 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 4145	. . . . 5 |- (/) e. _V
2	1	snid 3638	. . . 4 |- (/) e. { (/) }
3		ssrab2 3255	. . . . 5 |- { z e. { (/) } | ph } C_ { (/) }
4		p0ex 4203	. . . . . . 7 |- { (/) } e. _V
5	4	rabex 4162	. . . . . 6 |- { z e. { (/) } | ph } e. _V
6		sseq12 3195	. . . . . . 7 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( x C_ y <-> { z e. { (/) } | ph } C_ { (/) } ) )
7		simpl 109	. . . . . . . . 9 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> x = { z e. { (/) } | ph } )
8		simpr 110	. . . . . . . . . 10 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> y = { (/) } )
9	8, 7	difeq12d 3269	. . . . . . . . 9 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( y \ x ) = ( { (/) } \ { z e. { (/) } | ph } ) )
10	7, 9	uneq12d 3305	. . . . . . . 8 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( x u. ( y \ x ) ) = ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) )
11	10, 8	eqeq12d 2204	. . . . . . 7 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( ( x u. ( y \ x ) ) = y <-> ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) = { (/) } ) )
12	6, 11	bibi12d 235	. . . . . 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 2807	. . . . 5 |- ( { z e. { (/) } | ph } C_ { (/) } <-> ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) = { (/) } )
15	3, 14	mpbi 145	. . . 4 |- ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) = { (/) }
16	2, 15	eleqtrri 2265	. . 3 |- (/) e. ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) )
17		elun 3291	. . 3 |- ( (/) e. ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) <-> ( (/) e. { z e. { (/) } | ph } \/ (/) e. ( { (/) } \ { z e. { (/) } | ph } ) ) )
18	16, 17	mpbi 145	. 2 |- ( (/) e. { z e. { (/) } | ph } \/ (/) e. ( { (/) } \ { z e. { (/) } | ph } ) )
19		biidd 172	. . . . . 6 |- ( z = (/) -> ( ph <-> ph ) )
20	19	elrab3 2909	. . . . 5 |- ( (/) e. { (/) } -> ( (/) e. { z e. { (/) } | ph } <-> ph ) )
21	2, 20	ax-mp 5	. . . 4 |- ( (/) e. { z e. { (/) } | ph } <-> ph )
22	21	biimpi 120	. . 3 |- ( (/) e. { z e. { (/) } | ph } -> ph )
23		eldifn 3273	. . . 4 |- ( (/) e. ( { (/) } \ { z e. { (/) } | ph } ) -> -. (/) e. { z e. { (/) } | ph } )
24	23, 21	sylnib 677	. . 3 |- ( (/) e. ( { (/) } \ { z e. { (/) } | ph } ) -> -. ph )
25	22, 24	orim12i 760	. 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 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2160   {crab 2472   \ cdif 3141   u. cun 3142   C_ wss 3144   (/)c0 3437   {csn 3607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613

This theorem is referenced by: (None)