Theorem isinfcard 3692

Description: Two ways of expressing the property of being a transfinite cardinal.

Assertion

Ref	Expression
isinfcard	|- ((om (_ A /\ (card` A) = A) <-> A e. ran aleph)

Proof of Theorem isinfcard

Step	Hyp	Ref	Expression
1		cleqcom 1103	. . 3 |- ((aleph` x) = A <-> A = (aleph` x))
2	1	birex 1224	. 2 |- (E.x e. On (aleph` x) = A <-> E.x e. On A = (aleph` x))
3		alephfnon 3668	. . 3 |- aleph Fn On
4		fvelrn 2883	. . 3 |- (aleph Fn On -> (A e. ran aleph <-> E.x e. On (aleph` x) = A))
5	3, 4	ax-mp 6	. 2 |- (A e. ran aleph <-> E.x e. On (aleph` x) = A)
6		cardalephex 3691	. . . 4 |- (om (_ A -> ((card` A) = A <-> E.x e. On A = (aleph` x)))
7	6	pm5.32i 489	. . 3 |- ((om (_ A /\ (card` A) = A) <-> (om (_ A /\ E.x e. On A = (aleph` x)))
8		sseq2 1522	. . . . . . 7 |- (A = (aleph` x) -> (om (_ A <-> om (_ (aleph` x)))
9		alephgeom 3687	. . . . . . . 8 |- (x e. On <-> om (_ (aleph` x))
10	9	biimp 133	. . . . . . 7 |- (x e. On -> om (_ (aleph` x))
11	8, 10	syl5bir 184	. . . . . 6 |- (A = (aleph` x) -> (x e. On -> om (_ A))
12	11	com12 13	. . . . 5 |- (x e. On -> (A = (aleph` x) -> om (_ A))
13	12	r19.23aiv 1284	. . . 4 |- (E.x e. On A = (aleph` x) -> om (_ A)
14	13	pm4.71ri 484	. . 3 |- (E.x e. On A = (aleph` x) <-> (om (_ A /\ E.x e. On A = (aleph` x)))
15	7, 14	bitr4 154	. 2 |- ((om (_ A /\ (card` A) = A) <-> E.x e. On A = (aleph` x))
16	2, 5, 15	3bitr4r 159	1 |- ((om (_ A /\ (card` A) = A) <-> A e. ran aleph)

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  E.wrex 1202   (_ wss 1487  Oncon0 2199  omcom 2372  ran crn 2411   Fn wfn 2417  ` cfv 2422  cardccrd 3620  alephcale 3621

This theorem is referenced by:  iscard3 3693  cardinfima 3696  alephiso 3697  alephsson 3699

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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