Theorem exmidac 7271

Description: The axiom of choice implies excluded middle. See acexmid 5918 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)

Assertion

Ref	Expression
exmidac	|- (CHOICE -> EXMID )

Proof of Theorem exmidac

Dummy variables x u y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2200	. . . 4 |- ( u = x -> ( u = (/) <-> x = (/) ) )
2	1	orbi1d 792	. . 3 |- ( u = x -> ( ( u = (/) \/ y = { (/) } ) <-> ( x = (/) \/ y = { (/) } ) ) )
3	2	cbvrabv 2759	. 2 |- { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } = { x e. { (/) , { (/) } } | ( x = (/) \/ y = { (/) } ) }
4		eqeq1 2200	. . . 4 |- ( u = x -> ( u = { (/) } <-> x = { (/) } ) )
5	4	orbi1d 792	. . 3 |- ( u = x -> ( ( u = { (/) } \/ y = { (/) } ) <-> ( x = { (/) } \/ y = { (/) } ) ) )
6	5	cbvrabv 2759	. 2 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ y = { (/) } ) }
7		eqid 2193	. 2 |- { { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } , { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } } = { { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } , { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } }
8	3, 6, 7	exmidaclem 7270	1 |- (CHOICE -> EXMID )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   {crab 2476   (/)c0 3447   {csn 3619   {cpr 3620  EXMIDwem 4224  CHOICEwac 7267

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-exmid 4225  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ac 7268

This theorem is referenced by: (None)