Theorem undifexmid 4289

Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3577 and undifdcss 7158 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 4221	. . . . 5 |- (/) e. _V
2	1	snid 3704	. . . 4 |- (/) e. { (/) }
3		ssrab2 3313	. . . . 5 |- { z e. { (/) } | ph } C_ { (/) }
4		p0ex 4284	. . . . . . 7 |- { (/) } e. _V
5	4	rabex 4239	. . . . . 6 |- { z e. { (/) } | ph } e. _V
6		sseq12 3253	. . . . . . 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 3328	. . . . . . . . 9 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( y \ x ) = ( { (/) } \ { z e. { (/) } | ph } ) )
10	7, 9	uneq12d 3364	. . . . . . . 8 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( x u. ( y \ x ) ) = ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) ) )
11	10, 8	eqeq12d 2246	. . . . . . 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 2860	. . . . 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 2307	. . 3 |- (/) e. ( { z e. { (/) } | ph } u. ( { (/) } \ { z e. { (/) } | ph } ) )
17		elun 3350	. . 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 2964	. . . . 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 3332	. . . 4 |- ( (/) e. ( { (/) } \ { z e. { (/) } | ph } ) -> -. (/) e. { z e. { (/) } | ph } )
24	23, 21	sylnib 683	. . 3 |- ( (/) e. ( { (/) } \ { z e. { (/) } | ph } ) -> -. ph )
25	22, 24	orim12i 767	. 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 716   = wceq 1398   e. wcel 2202   {crab 2515   \ cdif 3198   u. cun 3199   C_ wss 3201   (/)c0 3496   {csn 3673

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679

This theorem is referenced by: (None)