Theorem iscard 4825

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 4799	. . . 4 |- (card` A) e. On
2		eleq1 1526	. . . 4 |- ((card` A) = A -> ((card` A) e. On <-> A e. On))
3	1, 2	mpbii 193	. . 3 |- ((card` A) = A -> A e. On)
4	3	pm4.71ri 636	. 2 |- ((card` A) = A <-> (A e. On /\ (card` A) = A))
5		cardonle 4794	. . . . 5 |- (A e. On -> (card` A) (_ A)
6		eqss 2067	. . . . . 6 |- ((card` A) = A <-> ((card` A) (_ A /\ A (_ (card` A)))
7	6	baibr 684	. . . . 5 |- ((card` A) (_ A -> (A (_ (card` A) <-> (card` A) = A))
8	5, 7	syl 10	. . . 4 |- (A e. On -> (A (_ (card` A) <-> (card` A) = A))
9		onelon 2962	. . . . . . 7 |- ((A e. On /\ x e. A) -> x e. On)
10		cardsdomel 4824	. . . . . . 7 |- (x e. On -> (x ~< A <-> x e. (card` A)))
11	9, 10	syl 10	. . . . . 6 |- ((A e. On /\ x e. A) -> (x ~< A <-> x e. (card` A)))
12	11	ralbidva 1651	. . . . 5 |- (A e. On -> (A.x e. A x ~< A <-> A.x e. A x e. (card` A)))
13		dfss3 2049	. . . . 5 |- (A (_ (card` A) <-> A.x e. A x e. (card` A))
14	12, 13	syl6rbbr 537	. . . 4 |- (A e. On -> (A (_ (card` A) <-> A.x e. A x ~< A))
15	8, 14	bitr3d 528	. . 3 |- (A e. On -> ((card` A) = A <-> A.x e. A x ~< A))
16	15	pm5.32i 643	. 2 |- ((A e. On /\ (card` A) = A) <-> (A e. On /\ A.x e. A x ~< A))
17	4, 16	bitr 173	1 |- ((card` A) = A <-> (A e. On /\ A.x e. A x ~< A))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637   (_ wss 2037   class class class wbr 2609  Oncon0 2938  ` cfv 3172   ~< csdm 4350  cardccrd 4785

This theorem is referenced by:  cardmin 4832

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788

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