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Theorem alephsson 8472
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 8464 . . 3  |-  ( ( om  C_  x  /\  ( card `  x )  =  x )  <->  x  e.  ran  aleph )
2 cardon 8316 . . . . 5  |-  ( card `  x )  e.  On
3 eleq1 2526 . . . . 5  |-  ( (
card `  x )  =  x  ->  ( (
card `  x )  e.  On  <->  x  e.  On ) )
42, 3mpbii 211 . . . 4  |-  ( (
card `  x )  =  x  ->  x  e.  On )
54adantl 464 . . 3  |-  ( ( om  C_  x  /\  ( card `  x )  =  x )  ->  x  e.  On )
61, 5sylbir 213 . 2  |-  ( x  e.  ran  aleph  ->  x  e.  On )
76ssriv 3493 1  |-  ran  aleph  C_  On
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   Oncon0 4867   ran crn 4989   ` cfv 5570   omcom 6673   cardccrd 8307   alephcale 8308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-har 7976  df-card 8311  df-aleph 8312
This theorem is referenced by:  unialeph  8473  alephsmo  8474  alephfplem3  8478  alephfp  8480  alephfp2  8481
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