Theorem 2omotap 7326

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 7325	. . . . 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 4241	1 |- ( E* r r TAp 2o -> EXMID )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 831   = wceq 1364   E*wmo 2046   C_ wss 3157   (/)c0 3450   {csn 3622  EXMIDwem 4227   2oc2o 6468   TAp wtap 7316

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-tr 4132  df-exmid 4228  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-1o 6474  df-2o 6475  df-pap 7315  df-tap 7317

This theorem is referenced by:  exmidmotap  7328