Theorem pwfi 5661

Description: The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105.

Assertion

Ref	Expression
pwfi	|- (A e. Fin <-> ~PA e. Fin)

Proof of Theorem pwfi

Step	Hyp	Ref	Expression
1		isfi 5441	. . 3 |- (A e. Fin <-> E.m e. om A ~~ m)
2		enfi 5627	. . . . . 6 |- ((~Pm e. _V /\ ~PA ~~ ~Pm) -> (~PA e. Fin <-> ~Pm e. Fin))
3		visset 2295	. . . . . . 7 |- m e. _V
4	3	pwex 3487	. . . . . 6 |- ~Pm e. _V
5	3	pwen 5597	. . . . . 6 |- (A ~~ m -> ~PA ~~ ~Pm)
6	2, 4, 5	sylancr 526	. . . . 5 |- (A ~~ m -> (~PA e. Fin <-> ~Pm e. Fin))
7		pweq 3036	. . . . . . 7 |- (m = (/) -> ~Pm = ~P(/))
8	7	eleq1d 1963	. . . . . 6 |- (m = (/) -> (~Pm e. Fin <-> ~P(/) e. Fin))
9		pweq 3036	. . . . . . 7 |- (m = k -> ~Pm = ~Pk)
10	9	eleq1d 1963	. . . . . 6 |- (m = k -> (~Pm e. Fin <-> ~Pk e. Fin))
11		pweq 3036	. . . . . . . 8 |- (m = suc k -> ~Pm = ~Psuc k)
12		df-suc 3663	. . . . . . . . 9 |- suc k = (k u. {k})
13		pweq 3036	. . . . . . . . 9 |- (suc k = (k u. {k}) -> ~Psuc k = ~P(k u. {k}))
14	12, 13	ax-mp 7	. . . . . . . 8 |- ~Psuc k = ~P(k u. {k})
15	11, 14	syl6eq 1944	. . . . . . 7 |- (m = suc k -> ~Pm = ~P(k u. {k}))
16	15	eleq1d 1963	. . . . . 6 |- (m = suc k -> (~Pm e. Fin <-> ~P(k u. {k}) e. Fin))
17		pw0 3132	. . . . . . . 8 |- ~P(/) = {(/)}
18		df1o2 5185	. . . . . . . 8 |- 1o = {(/)}
19	17, 18	eqtr4i 1911	. . . . . . 7 |- ~P(/) = 1o
20		1onn 5310	. . . . . . . 8 |- 1o e. om
21		ssid 2634	. . . . . . . 8 |- 1o C_ 1o
22		ssnnfi 5629	. . . . . . . 8 |- ((1o e. om /\ 1o C_ 1o) -> 1o e. Fin)
23	20, 21, 22	mp2an 761	. . . . . . 7 |- 1o e. Fin
24	19, 23	eqeltri 1967	. . . . . 6 |- ~P(/) e. Fin
25		eqid 1884	. . . . . . . 8 |- {<.c, y>. | (c e. ~Pk /\ y = (c u. {k}))} = {<.c, y>. | (c e. ~Pk /\ y = (c u. {k}))}
26	25	pwfilem 5660	. . . . . . 7 |- (~Pk e. Fin -> ~P(k u. {k}) e. Fin)
27	26	a1i 8	. . . . . 6 |- (k e. om -> (~Pk e. Fin -> ~P(k u. {k}) e. Fin))
28	8, 10, 16, 24, 27	finds1 3982	. . . . 5 |- (m e. om -> ~Pm e. Fin)
29	6, 28	syl5cbir 228	. . . 4 |- (m e. om -> (A ~~ m -> ~PA e. Fin))
30	29	r19.23aiv 2211	. . 3 |- (E.m e. om A ~~ m -> ~PA e. Fin)
31	1, 30	sylbi 216	. 2 |- (A e. Fin -> ~PA e. Fin)
32		elisset 2299	. . . . 5 |- (~PA e. Fin -> ~PA e. _V)
33		pwexb 3852	. . . . 5 |- (A e. _V <-> ~PA e. _V)
34	32, 33	sylibr 217	. . . 4 |- (~PA e. Fin -> A e. _V)
35		canth2g 5549	. . . 4 |- (A e. _V -> A ~< ~PA)
36		sdomdom 5445	. . . 4 |- (A ~< ~PA -> A ~<_ ~PA)
37	34, 35, 36	3syl 24	. . 3 |- (~PA e. Fin -> A ~<_ ~PA)
38		domfi 5631	. . 3 |- ((~PA e. Fin /\ A ~<_ ~PA) -> A e. Fin)
39	37, 38	mpdan 768	. 2 |- (~PA e. Fin -> A e. Fin)
40	31, 39	impbii 174	1 |- (A e. Fin <-> ~PA e. Fin)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  {csn 3044   class class class wbr 3338  {copab 3395  suc csuc 3659  omcom 3949  1oc1o 5172   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425  Fincfn 5426

This theorem is referenced by:  dominf 6052  unfinsef 14375  mapfi 15727  heiborlem18 15972  rrntotbnd 16022

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-2o 5178  df-oadd 5179  df-er 5318  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430

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