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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 7238. 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 5304 | . . 3 ⊢ 𝒫 𝐴 ∈ V |
3 | snelpwi 5361 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
4 | vex 3437 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | 4 | sneqr 4772 | . . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
6 | sneq 4572 | . . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
7 | 5, 6 | impbii 208 | . . . . 5 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦) |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) |
9 | 3, 8 | dom3 8793 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴) |
10 | 1, 2, 9 | mp2an 689 | . 2 ⊢ 𝐴 ≼ 𝒫 𝐴 |
11 | 1 | canth 7238 | . . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴 |
12 | f1ofo 6732 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝒫 𝐴 → 𝑓:𝐴–onto→𝒫 𝐴) | |
13 | 11, 12 | mto 196 | . . . 4 ⊢ ¬ 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
14 | 13 | nex 1803 | . . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
15 | bren 8752 | . . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴) | |
16 | 14, 15 | mtbir 323 | . 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴 |
17 | brsdom 8772 | . 2 ⊢ (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴)) | |
18 | 10, 16, 17 | mpbir2an 708 | 1 ⊢ 𝐴 ≺ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2107 Vcvv 3433 𝒫 cpw 4534 {csn 4562 class class class wbr 5075 –onto→wfo 6435 –1-1-onto→wf1o 6436 ≈ cen 8739 ≼ cdom 8740 ≺ csdm 8741 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-en 8743 df-dom 8744 df-sdom 8745 |
This theorem is referenced by: canth2g 8927 r1sdom 9541 alephsucpw2 9876 dfac13 9907 pwsdompw 9969 numthcor 10259 alephexp1 10344 pwcfsdom 10348 cfpwsdom 10349 gchac 10446 inawinalem 10454 tskcard 10546 gruina 10583 grothac 10595 rpnnen 15945 rexpen 15946 rucALT 15948 rectbntr0 24004 |
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