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Theorem alephord2 7700
Description: Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)
Assertion
Ref Expression
alephord2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  e.  ( aleph `  B )
) )

Proof of Theorem alephord2
StepHypRef Expression
1 alephord 7699 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
2 alephon 7693 . . . 4  |-  ( aleph `  A )  e.  On
3 alephon 7693 . . . . 5  |-  ( aleph `  B )  e.  On
4 onenon 7579 . . . . 5  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
53, 4ax-mp 10 . . . 4  |-  ( aleph `  B )  e.  dom  card
6 cardsdomel 7604 . . . 4  |-  ( ( ( aleph `  A )  e.  On  /\  ( aleph `  B )  e.  dom  card )  ->  ( ( aleph `  A )  ~< 
( aleph `  B )  <->  (
aleph `  A )  e.  ( card `  ( aleph `  B ) ) ) )
72, 5, 6mp2an 655 . . 3  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  <->  (
aleph `  A )  e.  ( card `  ( aleph `  B ) ) )
8 alephcard 7694 . . . 4  |-  ( card `  ( aleph `  B )
)  =  ( aleph `  B )
98eleq2i 2350 . . 3  |-  ( (
aleph `  A )  e.  ( card `  ( aleph `  B ) )  <-> 
( aleph `  A )  e.  ( aleph `  B )
)
107, 9bitri 242 . 2  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  <->  (
aleph `  A )  e.  ( aleph `  B )
)
111, 10syl6bb 254 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  e.  ( aleph `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1687   class class class wbr 4026   Oncon0 4393   dom cdm 4690   ` cfv 5223    ~< csdm 6859   cardccrd 7565   alephcale 7566
This theorem is referenced by:  alephord2i  7701  alephord3  7702  alephiso  7722  alephval3  7734
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-oi 7222  df-har 7269  df-card 7569  df-aleph 7570
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