Theorem exmidac 7065

Description: The axiom of choice implies excluded middle. See acexmid 5773 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 2146	. . . 4 |- ( u = x -> ( u = (/) <-> x = (/) ) )
2	1	orbi1d 780	. . 3 |- ( u = x -> ( ( u = (/) \/ y = { (/) } ) <-> ( x = (/) \/ y = { (/) } ) ) )
3	2	cbvrabv 2685	. 2 |- { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } = { x e. { (/) , { (/) } } | ( x = (/) \/ y = { (/) } ) }
4		eqeq1 2146	. . . 4 |- ( u = x -> ( u = { (/) } <-> x = { (/) } ) )
5	4	orbi1d 780	. . 3 |- ( u = x -> ( ( u = { (/) } \/ y = { (/) } ) <-> ( x = { (/) } \/ y = { (/) } ) ) )
6	5	cbvrabv 2685	. 2 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ y = { (/) } ) }
7		eqid 2139	. 2 |- { { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } , { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } } = { { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } , { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } }
8	3, 6, 7	exmidaclem 7064	1 |- (CHOICE -> EXMID )

Syntax hints:   -> wi 4   \/ wo 697   = wceq 1331   {crab 2420   (/)c0 3363   {csn 3527   {cpr 3528  EXMIDwem 4118  CHOICEwac 7061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-exmid 4119  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ac 7062

This theorem is referenced by: (None)