Theorem cardalephex 4886

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 (_ 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		fveq2 3724	. . . . . 6 |- (x = |^|{y e. On | A (_ (aleph` y)} -> (aleph` x) = (aleph` |^|{y e. On | A (_ (aleph` y)}))
2	1	eqeq2d 1486	. . . . 5 |- (x = |^|{y e. On | A (_ (aleph` y)} -> (A = (aleph` x) <-> A = (aleph` |^|{y e. On | A (_ (aleph` y)})))
3	2	rcla4ev 1877	. . . 4 |- ((|^|{y e. On | A (_ (aleph` y)} e. On /\ A = (aleph` |^|{y e. On | A (_ (aleph` y)})) -> E.x e. On A = (aleph` x))
4		pm3.26 319	. . . . 5 |- ((om (_ A /\ (card` A) = A) -> om (_ A)
5		cardaleph 4885	. . . . . . 7 |- ((om (_ A /\ (card` A) = A) -> A = (aleph` |^|{y e. On | A (_ (aleph` y)}))
6	5	sseq2d 2089	. . . . . 6 |- ((om (_ A /\ (card` A) = A) -> (om (_ A <-> om (_ (aleph` |^|{y e. On | A (_ (aleph` y)})))
7		alephgeom 4882	. . . . . 6 |- (|^|{y e. On | A (_ (aleph` y)} e. On <-> om (_ (aleph` |^|{y e. On | A (_ (aleph` y)}))
8	6, 7	syl6bbr 538	. . . . 5 |- ((om (_ A /\ (card` A) = A) -> (om (_ A <-> |^|{y e. On | A (_ (aleph` y)} e. On))
9	4, 8	mpbid 195	. . . 4 |- ((om (_ A /\ (card` A) = A) -> |^|{y e. On | A (_ (aleph` y)} e. On)
10	3, 9, 5	sylanc 471	. . 3 |- ((om (_ A /\ (card` A) = A) -> E.x e. On A = (aleph` x))
11	10	ex 373	. 2 |- (om (_ A -> ((card` A) = A -> E.x e. On A = (aleph` x)))
12		alephcard 4867	. . . . 5 |- (card` (aleph` x)) = (aleph` x)
13		fveq2 3724	. . . . 5 |- (A = (aleph` x) -> (card` A) = (card` (aleph` x)))
14		id 59	. . . . 5 |- (A = (aleph` x) -> A = (aleph` x))
15	12, 13, 14	3eqtr4a 1532	. . . 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 1743	. 2 |- (E.x e. On A = (aleph` x) -> (card` A) = A)
18	11, 17	impbid1 517	1 |- (om (_ A -> ((card` A) = A <-> E.x e. On A = (aleph` x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wrex 1646  {crab 1648   (_ wss 2047  |^|cint 2533  Oncon0 2948  omcom 3131  ` cfv 3182  cardccrd 4813  alephcale 4814

This theorem is referenced by:  isinfcard 4887  alephfp 4900  alephval2 4902  alephval3 4903

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-reg 4593  ax-inf2 4625  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-pss 2055  df-nul 2281  df-if 2362  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-lim 2953  df-suc 2954  df-om 3132  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-rdg 3932  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816  df-aleph 4817

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