Theorem iscard2 4826

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:   x,A

Proof of Theorem iscard2

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	. . . . . 6 |- (A e. On -> (card` A) (_ A)
6	5	biantrurd 725	. . . . 5 |- (A e. On -> (A (_ (card` A) <-> ((card` A) (_ A /\ A (_ (card` A))))
7		eqss 2067	. . . . 5 |- ((card` A) = A <-> ((card` A) (_ A /\ A (_ (card` A)))
8	6, 7	syl6rbbr 537	. . . 4 |- (A e. On -> ((card` A) = A <-> A (_ (card` A)))
9		ensymg 4392	. . . . . . . . . . . 12 |- (A e. On -> (x ~~ A -> A ~~ x))
10		visset 1804	. . . . . . . . . . . . 13 |- x e. V
11	10	ensym 4393	. . . . . . . . . . . 12 |- (A ~~ x -> x ~~ A)
12	9, 11	impbid1 515	. . . . . . . . . . 11 |- (A e. On -> (x ~~ A <-> A ~~ x))
13	12	anbi2d 614	. . . . . . . . . 10 |- (A e. On -> ((x e. On /\ x ~~ A) <-> (x e. On /\ A ~~ x)))
14		breq1 2612	. . . . . . . . . . 11 |- (y = x -> (y ~~ A <-> x ~~ A))
15	14	elrab 1896	. . . . . . . . . 10 |- (x e. {y e. On | y ~~ A} <-> (x e. On /\ x ~~ A))
16	13, 15	syl5bb 530	. . . . . . . . 9 |- (A e. On -> (x e. {y e. On | y ~~ A} <-> (x e. On /\ A ~~ x)))
17	16	imbi1d 611	. . . . . . . 8 |- (A e. On -> ((x e. {y e. On | y ~~ A} -> A (_ x) <-> ((x e. On /\ A ~~ x) -> A (_ x)))
18		impexp 347	. . . . . . . 8 |- (((x e. On /\ A ~~ x) -> A (_ x) <-> (x e. On -> (A ~~ x -> A (_ x)))
19	17, 18	syl6bb 534	. . . . . . 7 |- (A e. On -> ((x e. {y e. On | y ~~ A} -> A (_ x) <-> (x e. On -> (A ~~ x -> A (_ x))))
20	19	ralbidv2 1657	. . . . . 6 |- (A e. On -> (A.x e. {y e. On | y ~~ A}A (_ x <-> A.x e. On (A ~~ x -> A (_ x)))
21		ssint 2539	. . . . . 6 |- (A (_ |^|{y e. On | y ~~ A} <-> A.x e. {y e. On | y ~~ A}A (_ x)
22	20, 21	syl5bb 530	. . . . 5 |- (A e. On -> (A (_ |^|{y e. On | y ~~ A} <-> A.x e. On (A ~~ x -> A (_ x)))
23		cardval 4798	. . . . . 6 |- (card` A) = |^|{y e. On | y ~~ A}
24	23	sseq2i 2076	. . . . 5 |- (A (_ (card` A) <-> A (_ |^|{y e. On | y ~~ A})
25	22, 24	syl5bb 530	. . . 4 |- (A e. On -> (A (_ (card` A) <-> A.x e. On (A ~~ x -> A (_ x)))
26	8, 25	bitrd 526	. . 3 |- (A e. On -> ((card` A) = A <-> A.x e. On (A ~~ x -> A (_ x)))
27	26	pm5.32i 643	. 2 |- ((A e. On /\ (card` A) = A) <-> (A e. On /\ A.x e. On (A ~~ x -> A (_ x)))
28	4, 27	bitr 173	1 |- ((card` A) = A <-> (A e. On /\ A.x e. On (A ~~ x -> A (_ x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  {crab 1640   (_ wss 2037  |^|cint 2523   class class class wbr 2609  Oncon0 2938  ` cfv 3172   ~~ cen 4348  cardccrd 4785

This theorem is referenced by:  ondomcard 4829

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-card 4788

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