Theorem canth3 9421

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	⊢ (𝐴 ∈ 𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))

Proof of Theorem canth3

Step	Hyp	Ref	Expression
1		canth2g 8155	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴)
2		pwexg 4880	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
3		cardsdom 9415	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ V) → ((card‘𝐴) ∈ (card‘𝒫 𝐴) ↔ 𝐴 ≺ 𝒫 𝐴))
4	2, 3	mpdan 703	. 2 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ (card‘𝒫 𝐴) ↔ 𝐴 ≺ 𝒫 𝐴))
5	1, 4	mpbird 247	1 ⊢ (𝐴 ∈ 𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2030  Vcvv 3231  𝒫 cpw 4191   class class class wbr 4685  ‘cfv 5926   ≺ csdm 7996  cardccrd 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-ac2 9323

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-wrecs 7452  df-recs 7513  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803  df-ac 8977

This theorem is referenced by: (None)