Theorem exmidontri 7456

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 7439	. 2 |- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2		ontriexmidim 4620	. . . 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 4290	. 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 841   \/ w3o 1003   = wceq 1397   A.wral 2510   C_ wss 3200   (/)c0 3494   {csn 3669  EXMIDwem 4284   Oncon0 4460

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-uni 3894  df-tr 4188  df-exmid 4285  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by: (None)