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Theorem alephord2 7913
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 7912 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
2 alephon 7906 . . . 4  |-  ( aleph `  A )  e.  On
3 alephon 7906 . . . . 5  |-  ( aleph `  B )  e.  On
4 onenon 7792 . . . . 5  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
53, 4ax-mp 8 . . . 4  |-  ( aleph `  B )  e.  dom  card
6 cardsdomel 7817 . . . 4  |-  ( ( ( aleph `  A )  e.  On  /\  ( aleph `  B )  e.  dom  card )  ->  ( ( aleph `  A )  ~< 
( aleph `  B )  <->  (
aleph `  A )  e.  ( card `  ( aleph `  B ) ) ) )
72, 5, 6mp2an 654 . . 3  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  <->  (
aleph `  A )  e.  ( card `  ( aleph `  B ) ) )
8 alephcard 7907 . . . 4  |-  ( card `  ( aleph `  B )
)  =  ( aleph `  B )
98eleq2i 2468 . . 3  |-  ( (
aleph `  A )  e.  ( card `  ( aleph `  B ) )  <-> 
( aleph `  A )  e.  ( aleph `  B )
)
107, 9bitri 241 . 2  |-  ( (
aleph `  A )  ~< 
( aleph `  B )  <->  (
aleph `  A )  e.  ( aleph `  B )
)
111, 10syl6bb 253 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  (
aleph `  A )  e.  ( aleph `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   class class class wbr 4172   Oncon0 4541   dom cdm 4837   ` cfv 5413    ~< csdm 7067   cardccrd 7778   alephcale 7779
This theorem is referenced by:  alephord2i  7914  alephord3  7915  alephiso  7935  alephval3  7947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783
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