Theorem exmidac 7423

Description: The axiom of choice implies excluded middle. See acexmid 6016 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 2238	. . . 4 |- ( u = x -> ( u = (/) <-> x = (/) ) )
2	1	orbi1d 798	. . 3 |- ( u = x -> ( ( u = (/) \/ y = { (/) } ) <-> ( x = (/) \/ y = { (/) } ) ) )
3	2	cbvrabv 2801	. 2 |- { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } = { x e. { (/) , { (/) } } | ( x = (/) \/ y = { (/) } ) }
4		eqeq1 2238	. . . 4 |- ( u = x -> ( u = { (/) } <-> x = { (/) } ) )
5	4	orbi1d 798	. . 3 |- ( u = x -> ( ( u = { (/) } \/ y = { (/) } ) <-> ( x = { (/) } \/ y = { (/) } ) ) )
6	5	cbvrabv 2801	. 2 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ y = { (/) } ) }
7		eqid 2231	. 2 |- { { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } , { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } } = { { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } , { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } }
8	3, 6, 7	exmidaclem 7422	1 |- (CHOICE -> EXMID )

Syntax hints:   -> wi 4   \/ wo 715   = wceq 1397   {crab 2514   (/)c0 3494   {csn 3669   {cpr 3670  EXMIDwem 4284  CHOICEwac 7419

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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-exmid 4285  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ac 7420

This theorem is referenced by: (None)