Theorem iscard2 4854

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 4827	. . . 4 |- (card` A) e. On
2		eleq1 1534	. . . 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 638	. 2 |- ((card` A) = A <-> (A e. On /\ (card` A) = A))
5		cardonle 4822	. . . . . 6 |- (A e. On -> (card` A) (_ A)
6	5	biantrurd 727	. . . . 5 |- (A e. On -> (A (_ (card` A) <-> ((card` A) (_ A /\ A (_ (card` A))))
7		eqss 2077	. . . . 5 |- ((card` A) = A <-> ((card` A) (_ A /\ A (_ (card` A)))
8	6, 7	syl6rbbr 539	. . . 4 |- (A e. On -> ((card` A) = A <-> A (_ (card` A)))
9		ensymg 4411	. . . . . . . . . . . 12 |- (A e. On -> (x ~~ A -> A ~~ x))
10		visset 1813	. . . . . . . . . . . . 13 |- x e. V
11	10	ensym 4412	. . . . . . . . . . . 12 |- (A ~~ x -> x ~~ A)
12	9, 11	impbid1 517	. . . . . . . . . . 11 |- (A e. On -> (x ~~ A <-> A ~~ x))
13	12	anbi2d 616	. . . . . . . . . 10 |- (A e. On -> ((x e. On /\ x ~~ A) <-> (x e. On /\ A ~~ x)))
14		breq1 2622	. . . . . . . . . . 11 |- (y = x -> (y ~~ A <-> x ~~ A))
15	14	elrab 1905	. . . . . . . . . 10 |- (x e. {y e. On | y ~~ A} <-> (x e. On /\ x ~~ A))
16	13, 15	syl5bb 532	. . . . . . . . 9 |- (A e. On -> (x e. {y e. On | y ~~ A} <-> (x e. On /\ A ~~ x)))
17	16	imbi1d 613	. . . . . . . 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 536	. . . . . . 7 |- (A e. On -> ((x e. {y e. On | y ~~ A} -> A (_ x) <-> (x e. On -> (A ~~ x -> A (_ x))))
20	19	ralbidv2 1665	. . . . . 6 |- (A e. On -> (A.x e. {y e. On | y ~~ A}A (_ x <-> A.x e. On (A ~~ x -> A (_ x)))
21		ssint 2549	. . . . . 6 |- (A (_ |^|{y e. On | y ~~ A} <-> A.x e. {y e. On | y ~~ A}A (_ x)
22	20, 21	syl5bb 532	. . . . 5 |- (A e. On -> (A (_ |^|{y e. On | y ~~ A} <-> A.x e. On (A ~~ x -> A (_ x)))
23		cardval 4826	. . . . . 6 |- (card` A) = |^|{y e. On | y ~~ A}
24	23	sseq2i 2086	. . . . 5 |- (A (_ (card` A) <-> A (_ |^|{y e. On | y ~~ A})
25	22, 24	syl5bb 532	. . . 4 |- (A e. On -> (A (_ (card` A) <-> A.x e. On (A ~~ x -> A (_ x)))
26	8, 25	bitrd 528	. . 3 |- (A e. On -> ((card` A) = A <-> A.x e. On (A ~~ x -> A (_ x)))
27	26	pm5.32i 645	. 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 956   e. wcel 958  A.wral 1645  {crab 1648   (_ wss 2047  |^|cint 2533   class class class wbr 2619  Oncon0 2948  ` cfv 3182   ~~ cen 4364  cardccrd 4813

This theorem is referenced by:  ondomcard 4857

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-card 4816

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