Theorem exmidmotap 7323

Description: The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)

Assertion

Ref	Expression
exmidmotap	|- (EXMID <-> A. x E* r r TAp x )

Distinct variable group:   x, r

Proof of Theorem exmidmotap

Dummy variables s u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 529	. . . . . . . 8 |- ( (EXMID /\ ( r TAp x /\ s TAp x ) ) -> r TAp x )
2		exmidapne 7322	. . . . . . . . 9 |- (EXMID -> ( r TAp x <-> r = { <. u , v >. | ( ( u e. x /\ v e. x ) /\ u =/= v ) } ) )
3	2	adantr 276	. . . . . . . 8 |- ( (EXMID /\ ( r TAp x /\ s TAp x ) ) -> ( r TAp x <-> r = { <. u , v >. | ( ( u e. x /\ v e. x ) /\ u =/= v ) } ) )
4	1, 3	mpbid 147	. . . . . . 7 |- ( (EXMID /\ ( r TAp x /\ s TAp x ) ) -> r = { <. u , v >. | ( ( u e. x /\ v e. x ) /\ u =/= v ) } )
5		simprr 531	. . . . . . . 8 |- ( (EXMID /\ ( r TAp x /\ s TAp x ) ) -> s TAp x )
6		exmidapne 7322	. . . . . . . . 9 |- (EXMID -> ( s TAp x <-> s = { <. u , v >. | ( ( u e. x /\ v e. x ) /\ u =/= v ) } ) )
7	6	adantr 276	. . . . . . . 8 |- ( (EXMID /\ ( r TAp x /\ s TAp x ) ) -> ( s TAp x <-> s = { <. u , v >. | ( ( u e. x /\ v e. x ) /\ u =/= v ) } ) )
8	5, 7	mpbid 147	. . . . . . 7 |- ( (EXMID /\ ( r TAp x /\ s TAp x ) ) -> s = { <. u , v >. | ( ( u e. x /\ v e. x ) /\ u =/= v ) } )
9	4, 8	eqtr4d 2229	. . . . . 6 |- ( (EXMID /\ ( r TAp x /\ s TAp x ) ) -> r = s )
10	9	ex 115	. . . . 5 |- (EXMID -> ( ( r TAp x /\ s TAp x ) -> r = s ) )
11	10	alrimivv 1886	. . . 4 |- (EXMID -> A. r A. s ( ( r TAp x /\ s TAp x ) -> r = s ) )
12		tapeq1 7314	. . . . 5 |- ( r = s -> ( r TAp x <-> s TAp x ) )
13	12	mo4 2103	. . . 4 |- ( E* r r TAp x <-> A. r A. s ( ( r TAp x /\ s TAp x ) -> r = s ) )
14	11, 13	sylibr 134	. . 3 |- (EXMID -> E* r r TAp x )
15	14	alrimiv 1885	. 2 |- (EXMID -> A. x E* r r TAp x )
16		2onn 6576	. . . 4 |- 2o e. om
17		tapeq2 7315	. . . . . 6 |- ( x = 2o -> ( r TAp x <-> r TAp 2o ) )
18	17	mobidv 2078	. . . . 5 |- ( x = 2o -> ( E* r r TAp x <-> E* r r TAp 2o ) )
19	18	spcgv 2848	. . . 4 |- ( 2o e. om -> ( A. x E* r r TAp x -> E* r r TAp 2o ) )
20	16, 19	ax-mp 5	. . 3 |- ( A. x E* r r TAp x -> E* r r TAp 2o )
21		2omotap 7321	. . 3 |- ( E* r r TAp 2o -> EXMID )
22	20, 21	syl 14	. 2 |- ( A. x E* r r TAp x -> EXMID )
23	15, 22	impbii 126	1 |- (EXMID <-> A. x E* r r TAp x )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   E*wmo 2043   e. wcel 2164   =/= wne 2364   {copab 4090  EXMIDwem 4224   omcom 4623   2oc2o 6465   TAp wtap 7311

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-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-exmid 4225  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-2o 6472  df-pap 7310  df-tap 7312

This theorem is referenced by: (None)