Theorem exmidontri 7351

Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
exmidontri	|- (EXMID <-> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )

Distinct variable group:   x, y

Proof of Theorem exmidontri

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidontriim 7337	. 2 |- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2		ontriexmidim 4570	. . . 4 |- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> DECID z = { (/) } )
3	2	adantr 276	. . 3 |- ( ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) /\ z C_ { (/) } ) -> DECID z = { (/) } )
4	3	exmid1dc 4244	. 2 |- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> EXMID )
5	1, 4	impbii 126	1 |- (EXMID <-> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )

Syntax hints:   <-> wb 105  DECID wdc 836   \/ w3o 980   = wceq 1373   A.wral 2484   C_ wss 3166   (/)c0 3460   {csn 3633  EXMIDwem 4238   Oncon0 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-uni 3851  df-tr 4143  df-exmid 4239  df-iord 4413  df-on 4415  df-suc 4418

This theorem is referenced by: (None)