Theorem undifexmid 4223

Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3528 and undifdcss 6981 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 4157	. . . . 5 |- (/) e. _V
2	1	snid 3650	. . . 4 |- (/) e. { (/) }
3		ssrab2 3265	. . . . 5 |- { z e. { (/) } | ph } C_ { (/) }
4		p0ex 4218	. . . . . . 7 |- { (/) } e. _V
5	4	rabex 4174	. . . . . 6 |- { z e. { (/) } | ph } e. _V
6		sseq12 3205	. . . . . . 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 3279	. . . . . . . . 9 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( y \ x ) = ( { (/) } \ { z e. { (/) } | ph } ) )
10	7, 9	uneq12d 3315	. . . . . . . 8 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( x u. ( y \ x ) ) = ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) )
11	10, 8	eqeq12d 2208	. . . . . . 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 2816	. . . . 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 2269	. . 3 |- (/) e. ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) )
17		elun 3301	. . 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 2918	. . . . 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 3283	. . . 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 2164   {crab 2476   \ cdif 3151   u. cun 3152   C_ wss 3154   (/)c0 3447   {csn 3619

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 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625

This theorem is referenced by: (None)