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Mirrors > Home > MPE Home > Th. List > canth3 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
canth3 | ⊢ (𝐴 ∈ 𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴)) |
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)) |
Colors of variables: wff setvar class |
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) |
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