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Theorem canth3 9585
Description: Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)
Assertion
Ref Expression
canth3 (𝐴𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))

Proof of Theorem canth3
StepHypRef Expression
1 canth2g 8270 . 2 (𝐴𝑉𝐴 ≺ 𝒫 𝐴)
2 pwexg 4980 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
3 cardsdom 9579 . . 3 ((𝐴𝑉 ∧ 𝒫 𝐴 ∈ V) → ((card‘𝐴) ∈ (card‘𝒫 𝐴) ↔ 𝐴 ≺ 𝒫 𝐴))
42, 3mpdan 667 . 2 (𝐴𝑉 → ((card‘𝐴) ∈ (card‘𝒫 𝐴) ↔ 𝐴 ≺ 𝒫 𝐴))
51, 4mpbird 247 1 (𝐴𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2145  Vcvv 3351  𝒫 cpw 4297   class class class wbr 4786  cfv 6031  csdm 8108  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-ac2 9487
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-wrecs 7559  df-recs 7621  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-card 8965  df-ac 9139
This theorem is referenced by: (None)
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