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Theorem alephord2 7636
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 7635 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
2 alephon 7629 . . . 4  |-  ( aleph `  A )  e.  On
3 alephon 7629 . . . . 5  |-  ( aleph `  B )  e.  On
4 onenon 7515 . . . . 5  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
53, 4ax-mp 10 . . . 4  |-  ( aleph `  B )  e.  dom  card
6 cardsdomel 7540 . . . 4  |-  ( ( ( aleph `  A )  e.  On  /\  ( aleph `  B )  e.  dom  card )  ->  ( ( aleph `  A )  ~< 
( aleph `  B )  <->  (
aleph `  A )  e.  ( card `  ( aleph `  B ) ) ) )
72, 5, 6mp2an 656 . . 3  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  <->  (
aleph `  A )  e.  ( card `  ( aleph `  B ) ) )
8 alephcard 7630 . . . 4  |-  ( card `  ( aleph `  B )
)  =  ( aleph `  B )
98eleq2i 2320 . . 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 1621   class class class wbr 3963   Oncon0 4329   dom cdm 4626   ` cfv 4638    ~< csdm 6795   cardccrd 7501   alephcale 7502
This theorem is referenced by:  alephord2i  7637  alephord3  7638  alephiso  7658  alephval3  7670
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-har 7205  df-card 7505  df-aleph 7506
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