Theorem oncard 3636

Description: A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85.

Assertion

Ref	Expression
oncard	|- (E.x A = (card` x) <-> A = (card` A))

Distinct variable group(s):   x,A

Proof of Theorem oncard

Step	Hyp	Ref	Expression
1		cardid 3635	. . . . . . 7 |- (card` x) ~~ x
2		breq1 2065	. . . . . . 7 |- (A = (card` x) -> (A ~~ x <-> (card` x) ~~ x))
3	1, 2	mpbiri 169	. . . . . 6 |- (A = (card` x) -> A ~~ x)
4		cardid 3635	. . . . . . 7 |- (card` A) ~~ A
5		entrt 3319	. . . . . . 7 |- (((card` A) ~~ A /\ A ~~ x) -> (card` A) ~~ x)
6	4, 5	mpan 518	. . . . . 6 |- (A ~~ x -> (card` A) ~~ x)
7		cardon 3634	. . . . . . . 8 |- (card` A) e. On
8		breq1 2065	. . . . . . . . 9 |- (y = (card` A) -> (y ~~ x <-> (card` A) ~~ x))
9	8	onintss 2266	. . . . . . . 8 |- ((card` A) e. On -> ((card` A) ~~ x -> |^|{y e. On | y ~~ x} (_ (card` A)))
10	7, 9	ax-mp 6	. . . . . . 7 |- ((card` A) ~~ x -> |^|{y e. On | y ~~ x} (_ (card` A))
11		cardval 3633	. . . . . . 7 |- (card` x) = |^|{y e. On | y ~~ x}
12	10, 11	syl5ss 1544	. . . . . 6 |- ((card` A) ~~ x -> (card` x) (_ (card` A))
13	3, 6, 12	3syl 21	. . . . 5 |- (A = (card` x) -> (card` x) (_ (card` A))
14		sseq1 1521	. . . . 5 |- (A = (card` x) -> (A (_ (card` A) <-> (card` x) (_ (card` A)))
15	13, 14	mpbird 171	. . . 4 |- (A = (card` x) -> A (_ (card` A))
16		cardon 3634	. . . . . 6 |- (card` x) e. On
17		eleq1 1149	. . . . . 6 |- (A = (card` x) -> (A e. On <-> (card` x) e. On))
18	16, 17	mpbiri 169	. . . . 5 |- (A = (card` x) -> A e. On)
19		cardonle 3629	. . . . 5 |- (A e. On -> (card` A) (_ A)
20	18, 19	syl 12	. . . 4 |- (A = (card` x) -> (card` A) (_ A)
21	15, 20	eqssd 1518	. . 3 |- (A = (card` x) -> A = (card` A))
22	21	19.23aiv 952	. 2 |- (E.x A = (card` x) -> A = (card` A))
23		fvex 2838	. . . 4 |- (card` A) e. V
24		eleq1 1149	. . . 4 |- (A = (card` A) -> (A e. V <-> (card` A) e. V))
25	23, 24	mpbiri 169	. . 3 |- (A = (card` A) -> A e. V)
26		fveq2 2832	. . . . 5 |- (x = A -> (card` x) = (card` A))
27	26	cleq2d 1112	. . . 4 |- (x = A -> (A = (card` x) <-> A = (card` A)))
28	27	cla4egv 1397	. . 3 |- (A e. V -> (A = (card` A) -> E.x A = (card` x)))
29	25, 28	mpcom 49	. 2 |- (A = (card` A) -> E.x A = (card` x))
30	22, 29	impbi 139	1 |- (E.x A = (card` x) <-> A = (card` A))

Syntax hints:   -> wi 2   <-> wb 127  E.wex 678   = wceq 1091   e. wcel 1092  {crab 1204  Vcvv 1348   (_ wss 1487  |^|cint 1965   class class class wbr 2054  Oncon0 2199  ` cfv 2422   ~~ cen 3271  cardccrd 3620

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-er 3200  df-en 3274  df-card 3623

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