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Theorem unialeph 9526
 Description: The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
unialeph ran ℵ = On

Proof of Theorem unialeph
StepHypRef Expression
1 alephprc 9524 . . . 4 ¬ ran ℵ ∈ V
2 uniexb 7485 . . . 4 (ran ℵ ∈ V ↔ ran ℵ ∈ V)
31, 2mtbi 324 . . 3 ¬ ran ℵ ∈ V
4 elex 3512 . . 3 ( ran ℵ ∈ On → ran ℵ ∈ V)
53, 4mto 199 . 2 ¬ ran ℵ ∈ On
6 alephsson 9525 . . . 4 ran ℵ ⊆ On
7 ssorduni 7499 . . . 4 (ran ℵ ⊆ On → Ord ran ℵ)
86, 7ax-mp 5 . . 3 Ord ran ℵ
9 ordeleqon 7502 . . 3 (Ord ran ℵ ↔ ( ran ℵ ∈ On ∨ ran ℵ = On))
108, 9mpbi 232 . 2 ( ran ℵ ∈ On ∨ ran ℵ = On)
115, 10mtpor 1767 1 ran ℵ = On
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 843   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  ∪ cuni 4837  ran crn 5555  Ord word 6189  Oncon0 6190  ℵcale 9364 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-om 7580  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-oi 8973  df-har 9021  df-card 9367  df-aleph 9368 This theorem is referenced by: (None)
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