Theorem pwfi 4554

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	|- (E.n e. om A ~~ n <-> E.n e. om P~A ~~ n)

Distinct variable group:   A,n

Proof of Theorem pwfi

Step	Hyp	Ref	Expression
1		breq2 2619	. . . 4 |- (n = m -> (A ~~ n <-> A ~~ m))
2	1	cbvrexv 1798	. . 3 |- (E.n e. om A ~~ n <-> E.m e. om A ~~ m)
3		visset 1810	. . . . . . . 8 |- m e. V
4	3	pwen 4492	. . . . . . 7 |- (A ~~ m -> P~A ~~ P~m)
5	3	pwex 2741	. . . . . . . 8 |- P~m e. V
6		enen1 4466	. . . . . . . 8 |- ((P~m e. V /\ P~A ~~ P~m) -> (P~A ~~ n <-> P~m ~~ n))
7	5, 6	mpan 694	. . . . . . 7 |- (P~A ~~ P~m -> (P~A ~~ n <-> P~m ~~ n))
8	4, 7	syl 10	. . . . . 6 |- (A ~~ m -> (P~A ~~ n <-> P~m ~~ n))
9	8	rexbidv 1662	. . . . 5 |- (A ~~ m -> (E.n e. om P~A ~~ n <-> E.n e. om P~m ~~ n))
10		pweq 2400	. . . . . . . 8 |- (m = (/) -> P~m = P~(/))
11	10	breq1d 2625	. . . . . . 7 |- (m = (/) -> (P~m ~~ n <-> P~(/) ~~ n))
12	11	rexbidv 1662	. . . . . 6 |- (m = (/) -> (E.n e. om P~m ~~ n <-> E.n e. om P~(/) ~~ n))
13		pweq 2400	. . . . . . . 8 |- (m = k -> P~m = P~k)
14	13	breq1d 2625	. . . . . . 7 |- (m = k -> (P~m ~~ n <-> P~k ~~ n))
15	14	rexbidv 1662	. . . . . 6 |- (m = k -> (E.n e. om P~m ~~ n <-> E.n e. om P~k ~~ n))
16		df-suc 2950	. . . . . . . . . 10 |- suc k = (k u. {k})
17	16	eqeq2i 1483	. . . . . . . . 9 |- (m = suc k <-> m = (k u. {k}))
18		pweq 2400	. . . . . . . . 9 |- (m = (k u. {k}) -> P~m = P~(k u. {k}))
19	17, 18	sylbi 199	. . . . . . . 8 |- (m = suc k -> P~m = P~(k u. {k}))
20	19	breq1d 2625	. . . . . . 7 |- (m = suc k -> (P~m ~~ n <-> P~(k u. {k}) ~~ n))
21	20	rexbidv 1662	. . . . . 6 |- (m = suc k -> (E.n e. om P~m ~~ n <-> E.n e. om P~(k u. {k}) ~~ n))
22		1onn 4246	. . . . . . 7 |- 1o e. om
23		pw0 2465	. . . . . . . . 9 |- P~(/) = {(/)}
24		df1o2 4133	. . . . . . . . 9 |- 1o = {(/)}
25	23, 24	eqtr4 1496	. . . . . . . 8 |- P~(/) = 1o
26	22	elisseti 1815	. . . . . . . . 9 |- 1o e. V
27	26	enref 4381	. . . . . . . 8 |- 1o ~~ 1o
28	25, 27	eqbrtr 2630	. . . . . . 7 |- P~(/) ~~ 1o
29		breq2 2619	. . . . . . . 8 |- (n = 1o -> (P~(/) ~~ n <-> P~(/) ~~ 1o))
30	29	rcla4ev 1874	. . . . . . 7 |- ((1o e. om /\ P~(/) ~~ 1o) -> E.n e. om P~(/) ~~ n)
31	22, 28, 30	mp2an 696	. . . . . 6 |- E.n e. om P~(/) ~~ n
32		eqid 1474	. . . . . . . 8 |- {<.c, y>. | (c e. P~k /\ y = (c u. {k}))} = {<.c, y>. | (c e. P~k /\ y = (c u. {k}))}
33	32	pwfilem 4553	. . . . . . 7 |- (E.n e. om P~k ~~ n -> E.n e. om P~(k u. {k}) ~~ n)
34	33	a1i 8	. . . . . 6 |- (k e. om -> (E.n e. om P~k ~~ n -> E.n e. om P~(k u. {k}) ~~ n))
35	12, 15, 21, 31, 34	finds1 3155	. . . . 5 |- (m e. om -> E.n e. om P~m ~~ n)
36	9, 35	syl5cbir 211	. . . 4 |- (m e. om -> (A ~~ m -> E.n e. om P~A ~~ n))
37	36	r19.23aiv 1741	. . 3 |- (E.m e. om A ~~ m -> E.n e. om P~A ~~ n)
38	2, 37	sylbi 199	. 2 |- (E.n e. om A ~~ n -> E.n e. om P~A ~~ n)
39		relen 4363	. . . . . . . . 9 |- Rel ~~
40	39	brrelexi 3204	. . . . . . . 8 |- (P~A ~~ n -> P~A e. V)
41		pwexb 2904	. . . . . . . 8 |- (A e. V <-> P~A e. V)
42	40, 41	sylibr 200	. . . . . . 7 |- (P~A ~~ n -> A e. V)
43		canth2g 4474	. . . . . . 7 |- (A e. V -> A ~< P~A)
44	42, 43	syl 10	. . . . . 6 |- (P~A ~~ n -> A ~< P~A)
45	44	adantl 388	. . . . 5 |- ((n e. om /\ P~A ~~ n) -> A ~< P~A)
46	45	r19.23aiva 1742	. . . 4 |- (E.n e. om P~A ~~ n -> A ~< P~A)
47		sdomdom 4376	. . . 4 |- (A ~< P~A -> A ~<_ P~A)
48	46, 47	syl 10	. . 3 |- (E.n e. om P~A ~~ n -> A ~<_ P~A)
49		domfi 4525	. . 3 |- ((E.n e. om P~A ~~ n /\ A ~<_ P~A) -> E.n e. om A ~~ n)
50	48, 49	mpdan 703	. 2 |- (E.n e. om P~A ~~ n -> E.n e. om A ~~ n)
51	38, 50	impbi 157	1 |- (E.n e. om A ~~ n <-> E.n e. om P~A ~~ n)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  E.wrex 1644  Vcvv 1808   u. cun 2042  (/)c0 2277  P~cpw 2398  {csn 2406   class class class wbr 2615  {copab 2662  suc csuc 2946  omcom 3127  1oc1o 4121   ~~ cen 4357   ~<_ cdom 4358   ~< csdm 4359

This theorem is referenced by:  dominf 4887

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1o 4126  df-2o 4127  df-oadd 4128  df-er 4254  df-map 4317  df-en 4360  df-dom 4361  df-sdom 4362

Copyright terms: Public domain