Theorem exmidmotap 7447

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 7446	. . . . . . . . 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 7446	. . . . . . . . 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 2265	. . . . . 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 1921	. . . 4 |- (EXMID -> A. r A. s ( ( r TAp x /\ s TAp x ) -> r = s ) )
12		tapeq1 7438	. . . . 5 |- ( r = s -> ( r TAp x <-> s TAp x ) )
13	12	mo4 2139	. . . 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 1920	. 2 |- (EXMID -> A. x E* r r TAp x )
16		2onn 6667	. . . 4 |- 2o e. om
17		tapeq2 7439	. . . . . 6 |- ( x = 2o -> ( r TAp x <-> r TAp 2o ) )
18	17	mobidv 2113	. . . . 5 |- ( x = 2o -> ( E* r r TAp x <-> E* r r TAp 2o ) )
19	18	spcgv 2890	. . . 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 7445	. . 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 1393   = wceq 1395   E*wmo 2078   e. wcel 2200   =/= wne 2400   {copab 4144  EXMIDwem 4278   omcom 4682   2oc2o 6556   TAp wtap 7435

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-exmid 4279  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-1st 6286  df-2nd 6287  df-1o 6562  df-2o 6563  df-pap 7434  df-tap 7436

This theorem is referenced by: (None)