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Theorem unialeph 8868
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 8866 . . . 4 ¬ ran ℵ ∈ V
2 uniexb 6921 . . . 4 (ran ℵ ∈ V ↔ ran ℵ ∈ V)
31, 2mtbi 312 . . 3 ¬ ran ℵ ∈ V
4 elex 3198 . . 3 ( ran ℵ ∈ On → ran ℵ ∈ V)
53, 4mto 188 . 2 ¬ ran ℵ ∈ On
6 alephsson 8867 . . . 4 ran ℵ ⊆ On
7 ssorduni 6932 . . . 4 (ran ℵ ⊆ On → Ord ran ℵ)
86, 7ax-mp 5 . . 3 Ord ran ℵ
9 ordeleqon 6935 . . 3 (Ord ran ℵ ↔ ( ran ℵ ∈ On ∨ ran ℵ = On))
108, 9mpbi 220 . 2 ( ran ℵ ∈ On ∨ ran ℵ = On)
115, 10mtpor 1692 1 ran ℵ = On
Colors of variables: wff setvar class
Syntax hints:  wo 383   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555   cuni 4402  ran crn 5075  Ord word 5681  Oncon0 5682  cale 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-oi 8359  df-har 8407  df-card 8709  df-aleph 8710
This theorem is referenced by: (None)
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