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Mirrors > Home > MPE Home > Th. List > canth2 | Structured version Visualization version GIF version |
Description: Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 7384. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
canth2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
canth2 | ⊢ 𝐴 ≺ 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | canth2.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | pwex 5385 | . . 3 ⊢ 𝒫 𝐴 ∈ V |
3 | snelpwi 5453 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
4 | vex 3481 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | 4 | sneqr 4844 | . . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
6 | sneq 4640 | . . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
7 | 5, 6 | impbii 209 | . . . . 5 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦) |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) |
9 | 3, 8 | dom3 9034 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴) |
10 | 1, 2, 9 | mp2an 692 | . 2 ⊢ 𝐴 ≼ 𝒫 𝐴 |
11 | 1 | canth 7384 | . . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴 |
12 | f1ofo 6855 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝒫 𝐴 → 𝑓:𝐴–onto→𝒫 𝐴) | |
13 | 11, 12 | mto 197 | . . . 4 ⊢ ¬ 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
14 | 13 | nex 1796 | . . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
15 | bren 8993 | . . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴) | |
16 | 14, 15 | mtbir 323 | . 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴 |
17 | brsdom 9013 | . 2 ⊢ (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴)) | |
18 | 10, 16, 17 | mpbir2an 711 | 1 ⊢ 𝐴 ≺ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 𝒫 cpw 4604 {csn 4630 class class class wbr 5147 –onto→wfo 6560 –1-1-onto→wf1o 6561 ≈ cen 8980 ≼ cdom 8981 ≺ csdm 8982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-en 8984 df-dom 8985 df-sdom 8986 |
This theorem is referenced by: canth2g 9169 r1sdom 9811 alephsucpw2 10148 dfac13 10180 pwsdompw 10240 numthcor 10531 alephexp1 10616 pwcfsdom 10620 cfpwsdom 10621 gchac 10718 inawinalem 10726 tskcard 10818 gruina 10855 grothac 10867 rpnnen 16259 rexpen 16260 rucALT 16262 rectbntr0 24867 |
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