Theorem 2omotap 7353

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 7352	. . . . 5 |- ( ( E* r r TAp 2o /\ -. -. x = { (/) } ) -> x = { (/) } )
2	1	ex 115	. . . 4 |- ( E* r r TAp 2o -> ( -. -. x = { (/) } -> x = { (/) } ) )
3		df-stab 832	. . . 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 4251	1 |- ( E* r r TAp 2o -> EXMID )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 831   = wceq 1372   E*wmo 2054   C_ wss 3165   (/)c0 3459   {csn 3632  EXMIDwem 4237   2oc2o 6486   TAp wtap 7343

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-tr 4142  df-exmid 4238  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637  df-xp 4679  df-1o 6492  df-2o 6493  df-pap 7342  df-tap 7344

This theorem is referenced by:  exmidmotap  7355