Theorem oncard 4829

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 4828	. . . . . . 7 |- (card` x) ~~ x
2		breq1 2622	. . . . . . 7 |- (A = (card` x) -> (A ~~ x <-> (card` x) ~~ x))
3	1, 2	mpbiri 194	. . . . . 6 |- (A = (card` x) -> A ~~ x)
4		cardid 4828	. . . . . . 7 |- (card` A) ~~ A
5		entrt 4414	. . . . . . 7 |- (((card` A) ~~ A /\ A ~~ x) -> (card` A) ~~ x)
6	4, 5	mpan 695	. . . . . 6 |- (A ~~ x -> (card` A) ~~ x)
7		cardon 4827	. . . . . . . 8 |- (card` A) e. On
8		breq1 2622	. . . . . . . . 9 |- (y = (card` A) -> (y ~~ x <-> (card` A) ~~ x))
9	8	onintss 3011	. . . . . . . 8 |- ((card` A) e. On -> ((card` A) ~~ x -> |^|{y e. On | y ~~ x} (_ (card` A)))
10	7, 9	ax-mp 7	. . . . . . 7 |- ((card` A) ~~ x -> |^|{y e. On | y ~~ x} (_ (card` A))
11		cardval 4826	. . . . . . 7 |- (card` x) = |^|{y e. On | y ~~ x}
12	10, 11	syl5ss 2105	. . . . . 6 |- ((card` A) ~~ x -> (card` x) (_ (card` A))
13	3, 6, 12	3syl 20	. . . . 5 |- (A = (card` x) -> (card` x) (_ (card` A))
14		sseq1 2082	. . . . 5 |- (A = (card` x) -> (A (_ (card` A) <-> (card` x) (_ (card` A)))
15	13, 14	mpbird 196	. . . 4 |- (A = (card` x) -> A (_ (card` A))
16		cardon 4827	. . . . . 6 |- (card` x) e. On
17		eleq1 1534	. . . . . 6 |- (A = (card` x) -> (A e. On <-> (card` x) e. On))
18	16, 17	mpbiri 194	. . . . 5 |- (A = (card` x) -> A e. On)
19		cardonle 4822	. . . . 5 |- (A e. On -> (card` A) (_ A)
20	18, 19	syl 10	. . . 4 |- (A = (card` x) -> (card` A) (_ A)
21	15, 20	eqssd 2079	. . 3 |- (A = (card` x) -> A = (card` A))
22	21	19.23aiv 1295	. 2 |- (E.x A = (card` x) -> A = (card` A))
23		fvex 3732	. . . 4 |- (card` A) e. V
24		eleq1 1534	. . . 4 |- (A = (card` A) -> (A e. V <-> (card` A) e. V))
25	23, 24	mpbiri 194	. . 3 |- (A = (card` A) -> A e. V)
26		fveq2 3724	. . . . 5 |- (x = A -> (card` x) = (card` A))
27	26	eqeq2d 1486	. . . 4 |- (x = A -> (A = (card` x) <-> A = (card` A)))
28	27	cla4egv 1863	. . 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 157	1 |- (E.x A = (card` x) <-> A = (card` A))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  E.wex 980  {crab 1648  Vcvv 1811   (_ wss 2047  |^|cint 2533   class class class wbr 2619  Oncon0 2948  ` cfv 3182   ~~ cen 4364  cardccrd 4813

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-card 4816

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