Theorem 2omotap 7615

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 7614	. . . . 5 |- ( ( E* r r TAp 2o /\ -. -. x = { (/) } ) -> x = { (/) } )
2	1	ex 115	. . . 4 |- ( E* r r TAp 2o -> ( -. -. x = { (/) } -> x = { (/) } ) )
3		df-stab 843	. . . 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 4340	1 |- ( E* r r TAp 2o -> EXMID )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 842   = wceq 1402   E*wmo 2087   C_ wss 3220   (/)c0 3520   {csn 3705  EXMIDwem 4326   2oc2o 6671   TAp wtap 7604

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-tr 4225  df-exmid 4327  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-1o 6677  df-2o 6678  df-pap 7598  df-tap 7605

This theorem is referenced by:  exmidmotap  7617