Theorem exmidac 7210

Description: The axiom of choice implies excluded middle. See acexmid 5876 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 2184	. . . 4 |- ( u = x -> ( u = (/) <-> x = (/) ) )
2	1	orbi1d 791	. . 3 |- ( u = x -> ( ( u = (/) \/ y = { (/) } ) <-> ( x = (/) \/ y = { (/) } ) ) )
3	2	cbvrabv 2738	. 2 |- { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } = { x e. { (/) , { (/) } } | ( x = (/) \/ y = { (/) } ) }
4		eqeq1 2184	. . . 4 |- ( u = x -> ( u = { (/) } <-> x = { (/) } ) )
5	4	orbi1d 791	. . 3 |- ( u = x -> ( ( u = { (/) } \/ y = { (/) } ) <-> ( x = { (/) } \/ y = { (/) } ) ) )
6	5	cbvrabv 2738	. 2 |- { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } = { x e. { (/) , { (/) } } | ( x = { (/) } \/ y = { (/) } ) }
7		eqid 2177	. 2 |- { { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } , { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } } = { { u e. { (/) , { (/) } } | ( u = (/) \/ y = { (/) } ) } , { u e. { (/) , { (/) } } | ( u = { (/) } \/ y = { (/) } ) } }
8	3, 6, 7	exmidaclem 7209	1 |- (CHOICE -> EXMID )

Syntax hints:   -> wi 4   \/ wo 708   = wceq 1353   {crab 2459   (/)c0 3424   {csn 3594   {cpr 3595  EXMIDwem 4196  CHOICEwac 7206

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-exmid 4197  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ac 7207

This theorem is referenced by: (None)