Theorem iscard 3659

Description: Two ways to express the property of being a cardinal number.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem iscard

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	. . . . 5 |- (A e. On -> (card` A) (_ A)
6		eqss 1516	. . . . . 6 |- ((card` A) = A <-> ((card` A) (_ A /\ A (_ (card` A)))
7	6	baibr 507	. . . . 5 |- ((card` A) (_ A -> (A (_ (card` A) <-> (card` A) = A))
8	5, 7	syl 12	. . . 4 |- (A e. On -> (A (_ (card` A) <-> (card` A) = A))
9		onelon 2223	. . . . . . 7 |- ((A e. On /\ x e. A) -> x e. On)
10		cardsdomel 3658	. . . . . . 7 |- (x e. On -> (x ~< A <-> x e. (card` A)))
11	9, 10	syl 12	. . . . . 6 |- ((A e. On /\ x e. A) -> (x ~< A <-> x e. (card` A)))
12	11	biraldva 1215	. . . . 5 |- (A e. On -> (A.x e. A x ~< A <-> A.x e. A x e. (card` A)))
13		dfss3 1498	. . . . 5 |- (A (_ (card` A) <-> A.x e. A x e. (card` A))
14	12, 13	syl6rbbr 417	. . . 4 |- (A e. On -> (A (_ (card` A) <-> A.x e. A x ~< A))
15	8, 14	bitr3d 408	. . 3 |- (A e. On -> ((card` A) = A <-> A.x e. A x ~< A))
16	15	pm5.32i 489	. 2 |- ((A e. On /\ (card` A) = A) <-> (A e. On /\ A.x e. A x ~< A))
17	4, 16	bitr 151	1 |- ((card` A) = A <-> (A e. On /\ A.x e. A x ~< A))

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201   (_ wss 1487   class class class wbr 2054  Oncon0 2199  ` cfv 2422   ~< csdm 3273  cardccrd 3620

This theorem is referenced by:  cardmin 3666

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-dom 3275  df-sdom 3276  df-card 3623

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