Theorem pwf1oexmid 17029

Description: An exercise related to N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)

Hypothesis

Ref	Expression
pwle2.t	|- T = U_ x e. N ( { x } X. 1o )

Assertion

Ref	Expression
pwf1oexmid	|- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> ( ran G = ~P 1o <-> ( N = 2o /\ EXMID ) ) )

Distinct variable group:   x, N

Allowed substitution hints:   T( x)   G( x)

Proof of Theorem pwf1oexmid

Step	Hyp	Ref	Expression
1		pwle2.t	. . . . . 6 |- T = U_ x e. N ( { x } X. 1o )
2	1	pwle2 17028	. . . . 5 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> N C_ 2o )
3	2	adantr 276	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N C_ 2o )
4		pw1dom2 7586	. . . . . 6 |- 2o ~<_ ~P 1o
5		iunxpconst 4835	. . . . . . . . . . . 12 |- U_ x e. N ( { x } X. 1o ) = ( N X. 1o )
6		df1o2 6701	. . . . . . . . . . . . 13 |- 1o = { (/) }
7	6	xpeq2i 4795	. . . . . . . . . . . 12 |- ( N X. 1o ) = ( N X. { (/) } )
8	1, 5, 7	3eqtri 2263	. . . . . . . . . . 11 |- T = ( N X. { (/) } )
9		peano1 4741	. . . . . . . . . . . 12 |- (/) e. om
10		xpsneng 7120	. . . . . . . . . . . 12 |- ( ( N e. om /\ (/) e. om ) -> ( N X. { (/) } ) ~~ N )
11	9, 10	mpan2 429	. . . . . . . . . . 11 |- ( N e. om -> ( N X. { (/) } ) ~~ N )
12	8, 11	eqbrtrid 4165	. . . . . . . . . 10 |- ( N e. om -> T ~~ N )
13	12	ad2antrr 492	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T ~~ N )
14	13	ensymd 7070	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N ~~ T )
15		relen 7026	. . . . . . . . . 10 |- Rel ~~
16		brrelex1 4814	. . . . . . . . . 10 |- ( ( Rel ~~ /\ T ~~ N ) -> T e. _V )
17	15, 13, 16	sylancr 418	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T e. _V )
18		simplr 533	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> G : T -1-1-> ~P 1o )
19		simpr 110	. . . . . . . . . 10 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ran G = ~P 1o )
20		dff1o5 5648	. . . . . . . . . 10 |- ( G : T -1-1-onto-> ~P 1o <-> ( G : T -1-1-> ~P 1o /\ ran G = ~P 1o ) )
21	18, 19, 20	sylanbrc 421	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> G : T -1-1-onto-> ~P 1o )
22		f1oeng 7043	. . . . . . . . 9 |- ( ( T e. _V /\ G : T -1-1-onto-> ~P 1o ) -> T ~~ ~P 1o )
23	17, 21, 22	syl2anc 415	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> T ~~ ~P 1o )
24		entr 7071	. . . . . . . 8 |- ( ( N ~~ T /\ T ~~ ~P 1o ) -> N ~~ ~P 1o )
25	14, 23, 24	syl2anc 415	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N ~~ ~P 1o )
26	25	ensymd 7070	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ N )
27		domentr 7078	. . . . . 6 |- ( ( 2o ~<_ ~P 1o /\ ~P 1o ~~ N ) -> 2o ~<_ N )
28	4, 26, 27	sylancr 418	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> 2o ~<_ N )
29		2onn 6794	. . . . . . 7 |- 2o e. om
30		nndomo 7165	. . . . . . 7 |- ( ( 2o e. om /\ N e. om ) -> ( 2o ~<_ N <-> 2o C_ N ) )
31	29, 30	mpan 428	. . . . . 6 |- ( N e. om -> ( 2o ~<_ N <-> 2o C_ N ) )
32	31	ad2antrr 492	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ( 2o ~<_ N <-> 2o C_ N ) )
33	28, 32	mpbid 147	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> 2o C_ N )
34	3, 33	eqssd 3265	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> N = 2o )
35	26, 34	breqtrd 4156	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ~P 1o ~~ 2o )
36		exmidpw 7215	. . . 4 |- (EXMID <-> ~P 1o ~~ 2o )
37	35, 36	sylibr 134	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> EXMID )
38	34, 37	jca 306	. 2 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ran G = ~P 1o ) -> ( N = 2o /\ EXMID ) )
39		simplr 533	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> G : T -1-1-> ~P 1o )
40	12	ad2antrr 492	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ N )
41		simprl 535	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> N = 2o )
42	40, 41	breqtrd 4156	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ 2o )
43		simprr 537	. . . . . . . . 9 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> EXMID )
44	43, 36	sylib 122	. . . . . . . 8 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ~P 1o ~~ 2o )
45	44	ensymd 7070	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> 2o ~~ ~P 1o )
46		entr 7071	. . . . . . 7 |- ( ( T ~~ 2o /\ 2o ~~ ~P 1o ) -> T ~~ ~P 1o )
47	42, 45, 46	syl2anc 415	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> T ~~ ~P 1o )
48		nnfi 7174	. . . . . . . 8 |- ( 2o e. om -> 2o e. Fin )
49	29, 48	mp1i 10	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> 2o e. Fin )
50		enfi 7175	. . . . . . . 8 |- ( ~P 1o ~~ 2o -> ( ~P 1o e. Fin <-> 2o e. Fin ) )
51	44, 50	syl 14	. . . . . . 7 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ( ~P 1o e. Fin <-> 2o e. Fin ) )
52	49, 51	mpbird 167	. . . . . 6 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ~P 1o e. Fin )
53		f1finf1o 7264	. . . . . 6 |- ( ( T ~~ ~P 1o /\ ~P 1o e. Fin ) -> ( G : T -1-1-> ~P 1o <-> G : T -1-1-onto-> ~P 1o ) )
54	47, 52, 53	syl2anc 415	. . . . 5 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ( G : T -1-1-> ~P 1o <-> G : T -1-1-onto-> ~P 1o ) )
55	39, 54	mpbid 147	. . . 4 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> G : T -1-1-onto-> ~P 1o )
56	55, 20	sylib 122	. . 3 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ( G : T -1-1-> ~P 1o /\ ran G = ~P 1o ) )
57	56	simprd 114	. 2 |- ( ( ( N e. om /\ G : T -1-1-> ~P 1o ) /\ ( N = 2o /\ EXMID ) ) -> ran G = ~P 1o )
58	38, 57	impbida 604	1 |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> ( ran G = ~P 1o <-> ( N = 2o /\ EXMID ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   (/)c0 3520   ~Pcpw 3688   {csn 3709   U_ciun 4012   class class class wbr 4130  EXMIDwem 4331   omcom 4737   X. cxp 4772   ran crn 4775   Rel wrel 4779   -1-1->wf1 5374   -1-1-onto->wf1o 5376   1oc1o 6680   2oc2o 6681   ~~ cen 7020   ~<_ cdom 7021   Fincfn 7022

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-exmid 4332  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025

This theorem is used by: (None)