Theorem undifexmid 4283

Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3575 and undifdcss 7114 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 4216	. . . . 5 |- (/) e. _V
2	1	snid 3700	. . . 4 |- (/) e. { (/) }
3		ssrab2 3312	. . . . 5 |- { z e. { (/) } | ph } C_ { (/) }
4		p0ex 4278	. . . . . . 7 |- { (/) } e. _V
5	4	rabex 4234	. . . . . 6 |- { z e. { (/) } | ph } e. _V
6		sseq12 3252	. . . . . . 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 3326	. . . . . . . . 9 |- ( ( x = { z e. { (/) } | ph } /\ y = { (/) } ) -> ( y \ x ) = ( { (/) } \ { z e. { (/) } | ph } ) )
10	7, 9	uneq12d 3362	. . . . . . . 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 2859	. . . . 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 3348	. . 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 2963	. . . . 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 3330	. . . 4 |- ( (/) e. ( { (/) } \ { z e. { (/) } | ph } ) -> -. (/) e. { z e. { (/) } | ph } )
24	23, 21	sylnib 682	. . 3 |- ( (/) e. ( { (/) } \ { z e. { (/) } | ph } ) -> -. ph )
25	22, 24	orim12i 766	. 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 715   = wceq 1397   e. wcel 2202   {crab 2514   \ cdif 3197   u. cun 3198   C_ wss 3200   (/)c0 3494   {csn 3669

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675

This theorem is referenced by: (None)