Theorem canth3 10317

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 8918	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴)
2		pwexg 5301	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
3		cardsdom 10311	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ V) → ((card‘𝐴) ∈ (card‘𝒫 𝐴) ↔ 𝐴 ≺ 𝒫 𝐴))
4	2, 3	mpdan 684	. 2 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ (card‘𝒫 𝐴) ↔ 𝐴 ≺ 𝒫 𝐴))
5	1, 4	mpbird 256	1 ⊢ (𝐴 ∈ 𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  Vcvv 3432  𝒫 cpw 4533   class class class wbr 5074  ‘cfv 6433   ≺ csdm 8732  cardccrd 9693

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 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-ac2 10219

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-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  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 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-card 9697  df-ac 9872

This theorem is referenced by: (None)