Theorem exmidmotap 7408

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 7407	. . . . . . . . 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 7407	. . . . . . . . 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 2243	. . . . . 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 1899	. . . 4 |- (EXMID -> A. r A. s ( ( r TAp x /\ s TAp x ) -> r = s ) )
12		tapeq1 7399	. . . . 5 |- ( r = s -> ( r TAp x <-> s TAp x ) )
13	12	mo4 2117	. . . 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 1898	. 2 |- (EXMID -> A. x E* r r TAp x )
16		2onn 6630	. . . 4 |- 2o e. om
17		tapeq2 7400	. . . . . 6 |- ( x = 2o -> ( r TAp x <-> r TAp 2o ) )
18	17	mobidv 2091	. . . . 5 |- ( x = 2o -> ( E* r r TAp x <-> E* r r TAp 2o ) )
19	18	spcgv 2867	. . . 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 7406	. . 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 1371   = wceq 1373   E*wmo 2056   e. wcel 2178   =/= wne 2378   {copab 4120  EXMIDwem 4254   omcom 4656   2oc2o 6519   TAp wtap 7396

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-exmid 4255  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-1st 6249  df-2nd 6250  df-1o 6525  df-2o 6526  df-pap 7395  df-tap 7397

This theorem is referenced by: (None)