Theorem 2omotap 7260

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 7259	. . . . 5 |- ( ( E* r r TAp 2o /\ -. -. x = { (/) } ) -> x = { (/) } )
2	1	ex 115	. . . 4 |- ( E* r r TAp 2o -> ( -. -. x = { (/) } -> x = { (/) } ) )
3		df-stab 831	. . . 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 4210	1 |- ( E* r r TAp 2o -> EXMID )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 830   = wceq 1353   E*wmo 2027   C_ wss 3131   (/)c0 3424   {csn 3594  EXMIDwem 4196   2oc2o 6413   TAp wtap 7250

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-tr 4104  df-exmid 4197  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-1o 6419  df-2o 6420  df-pap 7249  df-tap 7251

This theorem is referenced by:  exmidmotap  7262