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Theorem alephprc 9502
 Description: The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
alephprc ¬ ran ℵ ∈ V

Proof of Theorem alephprc
StepHypRef Expression
1 cardprc 9385 . . . 4 {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V
21neli 3113 . . 3 ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V
3 cardnum 9497 . . . 4 {𝑥 ∣ (card‘𝑥) = 𝑥} = (ω ∪ ran ℵ)
43eleq1i 2902 . . 3 ({𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V ↔ (ω ∪ ran ℵ) ∈ V)
52, 4mtbi 325 . 2 ¬ (ω ∪ ran ℵ) ∈ V
6 omex 9082 . . 3 ω ∈ V
7 unexg 7447 . . 3 ((ω ∈ V ∧ ran ℵ ∈ V) → (ω ∪ ran ℵ) ∈ V)
86, 7mpan 689 . 2 (ran ℵ ∈ V → (ω ∪ ran ℵ) ∈ V)
95, 8mto 200 1 ¬ ran ℵ ∈ V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2115  {cab 2799  Vcvv 3471   ∪ cun 3908  ran crn 5529  ‘cfv 6328  ωcom 7555  cardccrd 9340  ℵcale 9341 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-oi 8950  df-har 8997  df-card 9344  df-aleph 9345 This theorem is referenced by:  unialeph  9504
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