Theorem pw1fin 7033

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 7032	. . . 4 |- (EXMID <-> ~P 1o = 2o )
2	1	biimpi 120	. . 3 |- (EXMID -> ~P 1o = 2o )
3		2onn 6630	. . . 4 |- 2o e. om
4		nnfi 6995	. . . 4 |- ( 2o e. om -> 2o e. Fin )
5	3, 4	ax-mp 5	. . 3 |- 2o e. Fin
6	2, 5	eqeltrdi 2298	. 2 |- (EXMID -> ~P 1o e. Fin )
7		df1o2 6538	. . . . . 6 |- 1o = { (/) }
8	7	sseq2i 3228	. . . . 5 |- ( x C_ 1o <-> x C_ { (/) } )
9		velpw 3633	. . . . . 6 |- ( x e. ~P 1o <-> x C_ 1o )
10		1oex 6533	. . . . . . . 8 |- 1o e. _V
11	10	pwid 3641	. . . . . . 7 |- 1o e. ~P 1o
12		fidceq 6992	. . . . . . 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 2218	. . . . 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 4260	. 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 2178   C_ wss 3174   (/)c0 3468   ~Pcpw 3626   {csn 3643  EXMIDwem 4254   omcom 4656   1oc1o 6518   2oc2o 6519   Fincfn 6850

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-tr 4159  df-exmid 4255  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1o 6525  df-2o 6526  df-en 6851  df-fin 6853

This theorem is referenced by: (None)