Theorem iscard 4864

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 4837	. . . 4 |- (card` A) e. On
2		eleq1 1537	. . . 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 640	. 2 |- ((card` A) = A <-> (A e. On /\ (card` A) = A))
5		cardonle 4832	. . . . 5 |- (A e. On -> (card` A) (_ A)
6		eqss 2080	. . . . . 6 |- ((card` A) = A <-> ((card` A) (_ A /\ A (_ (card` A)))
7	6	baibr 688	. . . . 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 2978	. . . . . . 7 |- ((A e. On /\ x e. A) -> x e. On)
10		cardsdomel 4863	. . . . . . 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 1662	. . . . 5 |- (A e. On -> (A.x e. A x ~< A <-> A.x e. A x e. (card` A)))
13		dfss3 2062	. . . . 5 |- (A (_ (card` A) <-> A.x e. A x e. (card` A))
14	12, 13	syl6rbbr 541	. . . 4 |- (A e. On -> (A (_ (card` A) <-> A.x e. A x ~< A))
15	8, 14	bitr3d 532	. . 3 |- (A e. On -> ((card` A) = A <-> A.x e. A x ~< A))
16	15	pm5.32i 647	. 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 958   e. wcel 960  A.wral 1648   (_ wss 2050   class class class wbr 2624  Oncon0 2954  ` cfv 3188   ~< csdm 4372  cardccrd 4823

This theorem is referenced by:  cardmin 4871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376  df-card 4826

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