Theorem 2omotap 7538

Description: If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotap	|- ( E* r r TAp 2o -> EXMID )

Proof of Theorem 2omotap

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2omotaplemst 7537	. . . . 5 |- ( ( E* r r TAp 2o /\ -. -. x = { (/) } ) -> x = { (/) } )
2	1	ex 115	. . . 4 |- ( E* r r TAp 2o -> ( -. -. x = { (/) } -> x = { (/) } ) )
3		df-stab 839	. . . 4 |- (STAB x = { (/) } <-> ( -. -. x = { (/) } -> x = { (/) } ) )
4	2, 3	sylibr 134	. . 3 |- ( E* r r TAp 2o -> STAB x = { (/) } )
5	4	adantr 276	. 2 |- ( ( E* r r TAp 2o /\ x C_ { (/) } ) -> STAB x = { (/) } )
6	5	exmid1stab 4304	1 |- ( E* r r TAp 2o -> EXMID )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 838   = wceq 1398   E*wmo 2080   C_ wss 3201   (/)c0 3496   {csn 3673  EXMIDwem 4290   2oc2o 6619   TAp wtap 7528

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  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  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-exmid 4291  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-1o 6625  df-2o 6626  df-pap 7527  df-tap 7529

This theorem is referenced by:  exmidmotap  7540