Theorem iscard 6005

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:   x,A

Proof of Theorem iscard

Step	Hyp	Ref	Expression
1		cardon 5976	. . . 4 |- (card` A) e. On
2		eleq1 1957	. . . 4 |- ((card` A) = A -> ((card` A) e. On <-> A e. On))
3	1, 2	mpbii 210	. . 3 |- ((card` A) = A -> A e. On)
4	3	pm4.71ri 700	. 2 |- ((card` A) = A <-> (A e. On /\ (card` A) = A))
5		cardonle 5868	. . . . 5 |- (A e. On -> (card` A) C_ A)
6		eqss 2631	. . . . . 6 |- ((card` A) = A <-> ((card` A) C_ A /\ A C_ (card` A)))
7	6	baibr 750	. . . . 5 |- ((card` A) C_ A -> (A C_ (card` A) <-> (card` A) = A))
8	5, 7	syl 12	. . . 4 |- (A e. On -> (A C_ (card` A) <-> (card` A) = A))
9		onelon 3683	. . . . . . 7 |- ((A e. On /\ x e. A) -> x e. On)
10		cardsdomel 6004	. . . . . . 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	ralbidva 2119	. . . . 5 |- (A e. On -> (A.x e. A x ~< A <-> A.x e. A x e. (card` A)))
13		dfss3 2611	. . . . 5 |- (A C_ (card` A) <-> A.x e. A x e. (card` A))
14	12, 13	syl6rbbr 598	. . . 4 |- (A e. On -> (A C_ (card` A) <-> A.x e. A x ~< A))
15	8, 14	bitr3d 589	. . 3 |- (A e. On -> ((card` A) = A <-> A.x e. A x ~< A))
16	15	pm5.32i 707	. 2 |- ((A e. On /\ (card` A) = A) <-> (A e. On /\ A.x e. A x ~< A))
17	4, 16	bitri 190	1 |- ((card` A) = A <-> (A e. On /\ A.x e. A x ~< A))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~< csdm 5425  cardccrd 5859

This theorem is referenced by:  cardmin 6012  carinttar 15279

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-dom 5428  df-sdom 5429  df-card 5862

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