Theorem oncard 5978

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:   x,A

Proof of Theorem oncard

Step	Hyp	Ref	Expression
1		cardid 5977	. . . . . . 7 |- (card` x) ~~ x
2		breq1 3341	. . . . . . 7 |- (A = (card` x) -> (A ~~ x <-> (card` x) ~~ x))
3	1, 2	mpbiri 211	. . . . . 6 |- (A = (card` x) -> A ~~ x)
4		cardid 5977	. . . . . . 7 |- (card` A) ~~ A
5		entr 5473	. . . . . . 7 |- (((card` A) ~~ A /\ A ~~ x) -> (card` A) ~~ x)
6	4, 5	mpan 759	. . . . . 6 |- (A ~~ x -> (card` A) ~~ x)
7		cardon 5976	. . . . . . . 8 |- (card` A) e. On
8		breq1 3341	. . . . . . . . 9 |- (y = (card` A) -> (y ~~ x <-> (card` A) ~~ x))
9	8	onintss 3713	. . . . . . . 8 |- ((card` A) e. On -> ((card` A) ~~ x -> |^|{y e. On | y ~~ x} C_ (card` A)))
10	7, 9	ax-mp 7	. . . . . . 7 |- ((card` A) ~~ x -> |^|{y e. On | y ~~ x} C_ (card` A))
11		cardval 5975	. . . . . . 7 |- (card` x) = |^|{y e. On | y ~~ x}
12	10, 11	syl5ss 2661	. . . . . 6 |- ((card` A) ~~ x -> (card` x) C_ (card` A))
13	3, 6, 12	3syl 24	. . . . 5 |- (A = (card` x) -> (card` x) C_ (card` A))
14		sseq1 2637	. . . . 5 |- (A = (card` x) -> (A C_ (card` A) <-> (card` x) C_ (card` A)))
15	13, 14	mpbird 213	. . . 4 |- (A = (card` x) -> A C_ (card` A))
16		cardon 5976	. . . . . 6 |- (card` x) e. On
17		eleq1 1957	. . . . . 6 |- (A = (card` x) -> (A e. On <-> (card` x) e. On))
18	16, 17	mpbiri 211	. . . . 5 |- (A = (card` x) -> A e. On)
19		cardonle 5868	. . . . 5 |- (A e. On -> (card` A) C_ A)
20	18, 19	syl 12	. . . 4 |- (A = (card` x) -> (card` A) C_ A)
21	15, 20	eqssd 2633	. . 3 |- (A = (card` x) -> A = (card` A))
22	21	19.23aiv 1674	. 2 |- (E.x A = (card` x) -> A = (card` A))
23		fvex 4689	. . . 4 |- (card` A) e. _V
24		eleq1 1957	. . . 4 |- (A = (card` A) -> (A e. _V <-> (card` A) e. _V))
25	23, 24	mpbiri 211	. . 3 |- (A = (card` A) -> A e. _V)
26		fveq2 4681	. . . . 5 |- (x = A -> (card` x) = (card` A))
27	26	eqeq2d 1895	. . . 4 |- (x = A -> (A = (card` x) <-> A = (card` A)))
28	27	cla4egv 2365	. . 3 |- (A e. _V -> (A = (card` A) -> E.x A = (card` x)))
29	25, 28	mpcom 60	. 2 |- (A = (card` A) -> E.x A = (card` x))
30	22, 29	impbii 174	1 |- (E.x A = (card` x) <-> A = (card` A))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  E.wex 1326  {crab 2108  _Vcvv 2292   C_ wss 2593  |^|cint 3214   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~~ cen 5423  cardccrd 5859

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  ax-ac 5906

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-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-suc 3663  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-er 5318  df-en 5427  df-card 5862

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