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Theorem aleph11 8546
Description: The aleph function is one-to-one. (Contributed by NM, 3-Aug-2004.)
Assertion
Ref Expression
aleph11  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  =  ( aleph `  B )  <->  A  =  B ) )

Proof of Theorem aleph11
StepHypRef Expression
1 alephord3 8540 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  C_  ( aleph `  B )
) )
2 alephord3 8540 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  C_  A  <->  (
aleph `  B )  C_  ( aleph `  A )
) )
32ancoms 459 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  A  <->  (
aleph `  B )  C_  ( aleph `  A )
) )
41, 3anbi12d 722 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  C_  B  /\  B  C_  A
)  <->  ( ( aleph `  A )  C_  ( aleph `  B )  /\  ( aleph `  B )  C_  ( aleph `  A )
) ) )
5 eqss 3459 . . 3  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
6 eqss 3459 . . 3  |-  ( (
aleph `  A )  =  ( aleph `  B )  <->  ( ( aleph `  A )  C_  ( aleph `  B )  /\  ( aleph `  B )  C_  ( aleph `  A )
) )
74, 5, 63bitr4g 296 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B  <-> 
( aleph `  A )  =  ( aleph `  B
) ) )
87bicomd 206 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A
)  =  ( aleph `  B )  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    C_ wss 3416   Oncon0 5446   ` cfv 5605   alephcale 8401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-om 6725  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-oi 8056  df-har 8104  df-card 8404  df-aleph 8405
This theorem is referenced by:  alephf1  8547  alephiso  8560
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