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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 7137. 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 5257 | . . 3 ⊢ 𝒫 𝐴 ∈ V |
3 | snelpwi 5313 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
4 | vex 3404 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | 4 | sneqr 4736 | . . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
6 | sneq 4536 | . . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
7 | 5, 6 | impbii 212 | . . . . 5 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦) |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) |
9 | 3, 8 | dom3 8612 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴) |
10 | 1, 2, 9 | mp2an 692 | . 2 ⊢ 𝐴 ≼ 𝒫 𝐴 |
11 | 1 | canth 7137 | . . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴 |
12 | f1ofo 6638 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝒫 𝐴 → 𝑓:𝐴–onto→𝒫 𝐴) | |
13 | 11, 12 | mto 200 | . . . 4 ⊢ ¬ 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
14 | 13 | nex 1807 | . . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
15 | bren 8577 | . . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴) | |
16 | 14, 15 | mtbir 326 | . 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴 |
17 | brsdom 8591 | . 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 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 Vcvv 3400 𝒫 cpw 4498 {csn 4526 class class class wbr 5040 –onto→wfo 6348 –1-1-onto→wf1o 6349 ≈ cen 8565 ≼ cdom 8566 ≺ csdm 8567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-en 8569 df-dom 8570 df-sdom 8571 |
This theorem is referenced by: canth2g 8734 r1sdom 9289 alephsucpw2 9624 dfac13 9655 pwsdompw 9717 numthcor 10007 alephexp1 10092 pwcfsdom 10096 cfpwsdom 10097 gchac 10194 inawinalem 10202 tskcard 10294 gruina 10331 grothac 10343 rpnnen 15685 rexpen 15686 rucALT 15688 rectbntr0 23597 |
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