Theorem pw1fin 7068

Description: Excluded middle is equivalent to the power set of 1o being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)

Assertion

Ref	Expression
pw1fin	|- (EXMID <-> ~P 1o e. Fin )

Proof of Theorem pw1fin

Step	Hyp	Ref	Expression
1		exmidpweq 7067	. . . 4 |- (EXMID <-> ~P 1o = 2o )
2	1	biimpi 120	. . 3 |- (EXMID -> ~P 1o = 2o )
3		2onn 6665	. . . 4 |- 2o e. om
4		nnfi 7030	. . . 4 |- ( 2o e. om -> 2o e. Fin )
5	3, 4	ax-mp 5	. . 3 |- 2o e. Fin
6	2, 5	eqeltrdi 2320	. 2 |- (EXMID -> ~P 1o e. Fin )
7		df1o2 6573	. . . . . 6 |- 1o = { (/) }
8	7	sseq2i 3251	. . . . 5 |- ( x C_ 1o <-> x C_ { (/) } )
9		velpw 3656	. . . . . 6 |- ( x e. ~P 1o <-> x C_ 1o )
10		1oex 6568	. . . . . . . 8 |- 1o e. _V
11	10	pwid 3664	. . . . . . 7 |- 1o e. ~P 1o
12		fidceq 7027	. . . . . . 7 |- ( ( ~P 1o e. Fin /\ x e. ~P 1o /\ 1o e. ~P 1o ) -> DECID x = 1o )
13	11, 12	mp3an3 1360	. . . . . 6 |- ( ( ~P 1o e. Fin /\ x e. ~P 1o ) -> DECID x = 1o )
14	9, 13	sylan2br 288	. . . . 5 |- ( ( ~P 1o e. Fin /\ x C_ 1o ) -> DECID x = 1o )
15	8, 14	sylan2br 288	. . . 4 |- ( ( ~P 1o e. Fin /\ x C_ { (/) } ) -> DECID x = 1o )
16	7	eqeq2i 2240	. . . . 5 |- ( x = 1o <-> x = { (/) } )
17	16	dcbii 845	. . . 4 |- (DECID x = 1o <-> DECID x = { (/) } )
18	15, 17	sylib 122	. . 3 |- ( ( ~P 1o e. Fin /\ x C_ { (/) } ) -> DECID x = { (/) } )
19	18	exmid1dc 4283	. 2 |- ( ~P 1o e. Fin -> EXMID )
20	6, 19	impbii 126	1 |- (EXMID <-> ~P 1o e. Fin )

Syntax hints:   /\ wa 104   <-> wb 105  DECID wdc 839   = wceq 1395   e. wcel 2200   C_ wss 3197   (/)c0 3491   ~Pcpw 3649   {csn 3666  EXMIDwem 4277   omcom 4681   1oc1o 6553   2oc2o 6554   Fincfn 6885

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-exmid 4278  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-2o 6561  df-en 6886  df-fin 6888

This theorem is referenced by: (None)