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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 6608. 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 4848 | . . 3 ⊢ 𝒫 𝐴 ∈ V |
3 | snelpwi 4912 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
4 | vex 3203 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | 4 | sneqr 4371 | . . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
6 | sneq 4187 | . . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
7 | 5, 6 | impbii 199 | . . . . 5 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦) |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) |
9 | 3, 8 | dom3 7999 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴) |
10 | 1, 2, 9 | mp2an 708 | . 2 ⊢ 𝐴 ≼ 𝒫 𝐴 |
11 | 1 | canth 6608 | . . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴 |
12 | f1ofo 6144 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝒫 𝐴 → 𝑓:𝐴–onto→𝒫 𝐴) | |
13 | 11, 12 | mto 188 | . . . 4 ⊢ ¬ 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
14 | 13 | nex 1731 | . . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
15 | bren 7964 | . . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴) | |
16 | 14, 15 | mtbir 313 | . 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴 |
17 | brsdom 7978 | . 2 ⊢ (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴)) | |
18 | 10, 16, 17 | mpbir2an 955 | 1 ⊢ 𝐴 ≺ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 𝒫 cpw 4158 {csn 4177 class class class wbr 4653 –onto→wfo 5886 –1-1-onto→wf1o 5887 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-en 7956 df-dom 7957 df-sdom 7958 |
This theorem is referenced by: canth2g 8114 r1sdom 8637 alephsucpw2 8934 dfac13 8964 pwsdompw 9026 numthcor 9316 alephexp1 9401 pwcfsdom 9405 cfpwsdom 9406 gchhar 9501 gchac 9503 inawinalem 9511 tskcard 9603 gruina 9640 grothac 9652 rpnnen 14956 rexpen 14957 rucALT 14959 rectbntr0 22635 |
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