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Theorem unialeph 9114
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 9112 . . . 4 ¬ ran ℵ ∈ V
2 uniexb 7138 . . . 4 (ran ℵ ∈ V ↔ ran ℵ ∈ V)
31, 2mtbi 311 . . 3 ¬ ran ℵ ∈ V
4 elex 3352 . . 3 ( ran ℵ ∈ On → ran ℵ ∈ V)
53, 4mto 188 . 2 ¬ ran ℵ ∈ On
6 alephsson 9113 . . . 4 ran ℵ ⊆ On
7 ssorduni 7150 . . . 4 (ran ℵ ⊆ On → Ord ran ℵ)
86, 7ax-mp 5 . . 3 Ord ran ℵ
9 ordeleqon 7153 . . 3 (Ord ran ℵ ↔ ( ran ℵ ∈ On ∨ ran ℵ = On))
108, 9mpbi 220 . 2 ( ran ℵ ∈ On ∨ ran ℵ = On)
115, 10mtpor 1844 1 ran ℵ = On
Colors of variables: wff setvar class
Syntax hints:  wo 382   = wceq 1632  wcel 2139  Vcvv 3340  wss 3715   cuni 4588  ran crn 5267  Ord word 5883  Oncon0 5884  cale 8952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-oi 8580  df-har 8628  df-card 8955  df-aleph 8956
This theorem is referenced by: (None)
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