Theorem iscard2 3660

Description: Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225.

Assertion

Ref	Expression
iscard2	|- ((card` A) = A <-> (A e. On /\ A.x e. On (A ~~ x -> A (_ x)))

Distinct variable group(s):   x,A

Proof of Theorem iscard2

Step	Hyp	Ref	Expression
1		cardon 3634	. . . 4 |- (card` A) e. On
2		eleq1 1149	. . . 4 |- ((card` A) = A -> ((card` A) e. On <-> A e. On))
3	1, 2	mpbii 168	. . 3 |- ((card` A) = A -> A e. On)
4	3	pm4.71ri 484	. 2 |- ((card` A) = A <-> (A e. On /\ (card` A) = A))
5		cardonle 3629	. . . . . 6 |- (A e. On -> (card` A) (_ A)
6	5	biantrurd 546	. . . . 5 |- (A e. On -> (A (_ (card` A) <-> ((card` A) (_ A /\ A (_ (card` A))))
7		eqss 1516	. . . . 5 |- ((card` A) = A <-> ((card` A) (_ A /\ A (_ (card` A)))
8	6, 7	syl6rbbr 417	. . . 4 |- (A e. On -> ((card` A) = A <-> A (_ (card` A)))
9		ensymg 3316	. . . . . . . . . . . 12 |- (A e. On -> (x ~~ A -> A ~~ x))
10		visset 1350	. . . . . . . . . . . . . 14 |- x e. V
11	10	ensym 3317	. . . . . . . . . . . . 13 |- (A ~~ x -> x ~~ A)
12	11	a1i 7	. . . . . . . . . . . 12 |- (A e. On -> (A ~~ x -> x ~~ A))
13	9, 12	impbid 397	. . . . . . . . . . 11 |- (A e. On -> (x ~~ A <-> A ~~ x))
14	13	anbi2d 468	. . . . . . . . . 10 |- (A e. On -> ((x e. On /\ x ~~ A) <-> (x e. On /\ A ~~ x)))
15		breq1 2065	. . . . . . . . . . 11 |- (y = x -> (y ~~ A <-> x ~~ A))
16	15	elrab 1422	. . . . . . . . . 10 |- (x e. {y e. On | y ~~ A} <-> (x e. On /\ x ~~ A))
17	14, 16	syl5bb 410	. . . . . . . . 9 |- (A e. On -> (x e. {y e. On | y ~~ A} <-> (x e. On /\ A ~~ x)))
18	17	imbi1d 465	. . . . . . . 8 |- (A e. On -> ((x e. {y e. On | y ~~ A} -> A (_ x) <-> ((x e. On /\ A ~~ x) -> A (_ x)))
19		impexp 276	. . . . . . . 8 |- (((x e. On /\ A ~~ x) -> A (_ x) <-> (x e. On -> (A ~~ x -> A (_ x)))
20	18, 19	syl6bb 414	. . . . . . 7 |- (A e. On -> ((x e. {y e. On | y ~~ A} -> A (_ x) <-> (x e. On -> (A ~~ x -> A (_ x))))
21	20	biraldv2 1221	. . . . . 6 |- (A e. On -> (A.x e. {y e. On | y ~~ A}A (_ x <-> A.x e. On (A ~~ x -> A (_ x)))
22		ssint 1980	. . . . . 6 |- (A (_ |^|{y e. On | y ~~ A} <-> A.x e. {y e. On | y ~~ A}A (_ x)
23	21, 22	syl5bb 410	. . . . 5 |- (A e. On -> (A (_ |^|{y e. On | y ~~ A} <-> A.x e. On (A ~~ x -> A (_ x)))
24		cardval 3633	. . . . . 6 |- (card` A) = |^|{y e. On | y ~~ A}
25	24	sseq2i 1525	. . . . 5 |- (A (_ (card` A) <-> A (_ |^|{y e. On | y ~~ A})
26	23, 25	syl5bb 410	. . . 4 |- (A e. On -> (A (_ (card` A) <-> A.x e. On (A ~~ x -> A (_ x)))
27	8, 26	bitrd 406	. . 3 |- (A e. On -> ((card` A) = A <-> A.x e. On (A ~~ x -> A (_ x)))
28	27	pm5.32i 489	. 2 |- ((A e. On /\ (card` A) = A) <-> (A e. On /\ A.x e. On (A ~~ x -> A (_ x)))
29	4, 28	bitr 151	1 |- ((card` A) = A <-> (A e. On /\ A.x e. On (A ~~ x -> A (_ x)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204   (_ wss 1487  |^|cint 1965   class class class wbr 2054  Oncon0 2199  ` cfv 2422   ~~ cen 3271  cardccrd 3620

This theorem is referenced by:  ondomcard 3663

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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