Theorem cardalephex 4866

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 3715	. . . . . 6 |- (x = |^|{y e. On | A (_ (aleph` y)} -> (aleph` x) = (aleph` |^|{y e. On | A (_ (aleph` y)}))
2	1	eqeq2d 1483	. . . . 5 |- (x = |^|{y e. On | A (_ (aleph` y)} -> (A = (aleph` x) <-> A = (aleph` |^|{y e. On | A (_ (aleph` y)})))
3	2	rcla4ev 1873	. . . 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 4865	. . . . . . 7 |- ((om (_ A /\ (card` A) = A) -> A = (aleph` |^|{y e. On | A (_ (aleph` y)}))
6	5	sseq2d 2085	. . . . . 6 |- ((om (_ A /\ (card` A) = A) -> (om (_ A <-> om (_ (aleph` |^|{y e. On | A (_ (aleph` y)})))
7		alephgeom 4862	. . . . . 6 |- (|^|{y e. On | A (_ (aleph` y)} e. On <-> om (_ (aleph` |^|{y e. On | A (_ (aleph` y)}))
8	6, 7	syl6bbr 537	. . . . 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 4847	. . . . 5 |- (card` (aleph` x)) = (aleph` x)
13		fveq2 3715	. . . . 5 |- (A = (aleph` x) -> (card` A) = (card` (aleph` x)))
14		id 59	. . . . 5 |- (A = (aleph` x) -> A = (aleph` x))
15	12, 13, 14	3eqtr4a 1529	. . . 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 1740	. 2 |- (E.x e. On A = (aleph` x) -> (card` A) = A)
18	11, 17	impbid1 516	1 |- (om (_ A -> ((card` A) = A <-> E.x e. On A = (aleph` x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wrex 1643  {crab 1645   (_ wss 2043  |^|cint 2528  Oncon0 2943  omcom 3126  ` cfv 3177  cardccrd 4793  alephcale 4794

This theorem is referenced by:  isinfcard 4867  alephfp 4880  alephval2 4882  alephval3 4883

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359  df-card 4796  df-aleph 4797

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