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Theorem alephsson 8291
Description: The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
alephsson  |-  ran  aleph  C_  On

Proof of Theorem alephsson
StepHypRef Expression
1 isinfcard 8283 . . 3  |-  ( ( om  C_  x  /\  ( card `  x )  =  x )  <->  x  e.  ran  aleph )
2 cardon 8135 . . . . 5  |-  ( card `  x )  e.  On
3 eleq1 2503 . . . . 5  |-  ( (
card `  x )  =  x  ->  ( (
card `  x )  e.  On  <->  x  e.  On ) )
42, 3mpbii 211 . . . 4  |-  ( (
card `  x )  =  x  ->  x  e.  On )
54adantl 466 . . 3  |-  ( ( om  C_  x  /\  ( card `  x )  =  x )  ->  x  e.  On )
61, 5sylbir 213 . 2  |-  ( x  e.  ran  aleph  ->  x  e.  On )
76ssriv 3381 1  |-  ran  aleph  C_  On
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3349   Oncon0 4740   ran crn 4862   ` cfv 5439   omcom 6497   cardccrd 8126   alephcale 8127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-om 6498  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-oi 7745  df-har 7794  df-card 8130  df-aleph 8131
This theorem is referenced by:  unialeph  8292  alephsmo  8293  alephfplem3  8297  alephfp  8299  alephfp2  8300
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