MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unialeph Structured version   Visualization version   GIF version

Theorem unialeph 9867
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 9865 . . . 4 ¬ ran ℵ ∈ V
2 uniexb 7604 . . . 4 (ran ℵ ∈ V ↔ ran ℵ ∈ V)
31, 2mtbi 322 . . 3 ¬ ran ℵ ∈ V
4 elex 3447 . . 3 ( ran ℵ ∈ On → ran ℵ ∈ V)
53, 4mto 196 . 2 ¬ ran ℵ ∈ On
6 alephsson 9866 . . . 4 ran ℵ ⊆ On
7 ssorduni 7619 . . . 4 (ran ℵ ⊆ On → Ord ran ℵ)
86, 7ax-mp 5 . . 3 Ord ran ℵ
9 ordeleqon 7622 . . 3 (Ord ran ℵ ↔ ( ran ℵ ∈ On ∨ ran ℵ = On))
108, 9mpbi 229 . 2 ( ran ℵ ∈ On ∨ ran ℵ = On)
115, 10mtpor 1773 1 ran ℵ = On
Colors of variables: wff setvar class
Syntax hints:  wo 844   = wceq 1539  wcel 2106  Vcvv 3429  wss 3886   cuni 4839  ran crn 5585  Ord word 6258  Oncon0 6259  cale 9704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-inf2 9386
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-se 5540  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-isom 6435  df-riota 7224  df-ov 7270  df-om 7703  df-2nd 7821  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-1o 8284  df-er 8485  df-en 8721  df-dom 8722  df-sdom 8723  df-fin 8724  df-oi 9256  df-har 9303  df-card 9707  df-aleph 9708
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator