Theorem cardalephex 6034

Description: Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse.

Assertion

Ref	Expression
cardalephex	|- (om C_ A -> ((card` A) = A <-> E.x e. On A = (aleph` x)))

Distinct variable group:   x,A

Proof of Theorem cardalephex

Step	Hyp	Ref	Expression
1		simpl 346	. . . . 5 |- ((om C_ A /\ (card` A) = A) -> om C_ A)
2		cardaleph 6033	. . . . . . 7 |- ((om C_ A /\ (card` A) = A) -> A = (aleph` |^|{y e. On | A C_ (aleph` y)}))
3	2	sseq2d 2645	. . . . . 6 |- ((om C_ A /\ (card` A) = A) -> (om C_ A <-> om C_ (aleph` |^|{y e. On | A C_ (aleph` y)})))
4		alephgeom 6030	. . . . . 6 |- (|^|{y e. On | A C_ (aleph` y)} e. On <-> om C_ (aleph` |^|{y e. On | A C_ (aleph` y)}))
5	3, 4	syl6bbr 597	. . . . 5 |- ((om C_ A /\ (card` A) = A) -> (om C_ A <-> |^|{y e. On | A C_ (aleph` y)} e. On))
6	1, 5	mpbid 212	. . . 4 |- ((om C_ A /\ (card` A) = A) -> |^|{y e. On | A C_ (aleph` y)} e. On)
7		fveq2 4681	. . . . . 6 |- (x = |^|{y e. On | A C_ (aleph` y)} -> (aleph` x) = (aleph` |^|{y e. On | A C_ (aleph` y)}))
8	7	eqeq2d 1895	. . . . 5 |- (x = |^|{y e. On | A C_ (aleph` y)} -> (A = (aleph` x) <-> A = (aleph` |^|{y e. On | A C_ (aleph` y)})))
9	8	rcla4ev 2381	. . . 4 |- ((|^|{y e. On | A C_ (aleph` y)} e. On /\ A = (aleph` |^|{y e. On | A C_ (aleph` y)})) -> E.x e. On A = (aleph` x))
10	6, 2, 9	syl11anc 524	. . 3 |- ((om C_ A /\ (card` A) = A) -> E.x e. On A = (aleph` x))
11	10	ex 402	. 2 |- (om C_ A -> ((card` A) = A -> E.x e. On A = (aleph` x)))
12		alephcard 6015	. . . . 5 |- (card` (aleph` x)) = (aleph` x)
13		fveq2 4681	. . . . 5 |- (A = (aleph` x) -> (card` A) = (card` (aleph` x)))
14		id 73	. . . . 5 |- (A = (aleph` x) -> A = (aleph` x))
15	12, 13, 14	3eqtr4a 1954	. . . 4 |- (A = (aleph` x) -> (card` A) = A)
16	15	a1i 8	. . 3 |- (x e. On -> (A = (aleph` x) -> (card` A) = A))
17	16	r19.23aiv 2211	. 2 |- (E.x e. On A = (aleph` x) -> (card` A) = A)
18	11, 17	impbid1 575	1 |- (om C_ A -> ((card` A) = A <-> E.x e. On A = (aleph` x)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  {crab 2108   C_ wss 2593  |^|cint 3214  Oncon0 3657  omcom 3949  ` cfv 3998  cardccrd 5859  alephcale 5860

This theorem is referenced by:  isinfcard 6035  alephfp 6048  alephval2 6050  alephval3 6051

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862  df-aleph 5863

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