Theorem canth3 9705

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 8389	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴)
2		pwexg 5080	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
3		cardsdom 9699	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ V) → ((card‘𝐴) ∈ (card‘𝒫 𝐴) ↔ 𝐴 ≺ 𝒫 𝐴))
4	2, 3	mpdan 678	. 2 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ (card‘𝒫 𝐴) ↔ 𝐴 ≺ 𝒫 𝐴))
5	1, 4	mpbird 249	1 ⊢ (𝐴 ∈ 𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))

Syntax hints:   → wi 4   ↔ wb 198   ∈ wcel 2164  Vcvv 3414  𝒫 cpw 4380   class class class wbr 4875  ‘cfv 6127   ≺ csdm 8227  cardccrd 9081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-ac2 9607

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-wrecs 7677  df-recs 7739  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-card 9085  df-ac 9259

This theorem is referenced by: (None)