Theorem pw1fin 7007

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 7006	. . . 4 |- (EXMID <-> ~P 1o = 2o )
2	1	biimpi 120	. . 3 |- (EXMID -> ~P 1o = 2o )
3		2onn 6607	. . . 4 |- 2o e. om
4		nnfi 6969	. . . 4 |- ( 2o e. om -> 2o e. Fin )
5	3, 4	ax-mp 5	. . 3 |- 2o e. Fin
6	2, 5	eqeltrdi 2296	. 2 |- (EXMID -> ~P 1o e. Fin )
7		df1o2 6515	. . . . . 6 |- 1o = { (/) }
8	7	sseq2i 3220	. . . . 5 |- ( x C_ 1o <-> x C_ { (/) } )
9		velpw 3623	. . . . . 6 |- ( x e. ~P 1o <-> x C_ 1o )
10		1oex 6510	. . . . . . . 8 |- 1o e. _V
11	10	pwid 3631	. . . . . . 7 |- 1o e. ~P 1o
12		fidceq 6966	. . . . . . 7 |- ( ( ~P 1o e. Fin /\ x e. ~P 1o /\ 1o e. ~P 1o ) -> DECID x = 1o )
13	11, 12	mp3an3 1339	. . . . . 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 2216	. . . . 5 |- ( x = 1o <-> x = { (/) } )
17	16	dcbii 842	. . . 4 |- (DECID x = 1o <-> DECID x = { (/) } )
18	15, 17	sylib 122	. . 3 |- ( ( ~P 1o e. Fin /\ x C_ { (/) } ) -> DECID x = { (/) } )
19	18	exmid1dc 4244	. 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 836   = wceq 1373   e. wcel 2176   C_ wss 3166   (/)c0 3460   ~Pcpw 3616   {csn 3633  EXMIDwem 4238   omcom 4638   1oc1o 6495   2oc2o 6496   Fincfn 6827

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-tr 4143  df-exmid 4239  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6502  df-2o 6503  df-en 6828  df-fin 6830

This theorem is referenced by: (None)