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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 6508. 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 4774 | . . 3 ⊢ 𝒫 𝐴 ∈ V |
3 | snelpwi 4839 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
4 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | 4 | sneqr 4311 | . . . . . 6 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
6 | sneq 4135 | . . . . . 6 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
7 | 5, 6 | impbii 198 | . . . . 5 ⊢ ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦) |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) |
9 | 3, 8 | dom3 7885 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴) |
10 | 1, 2, 9 | mp2an 704 | . 2 ⊢ 𝐴 ≼ 𝒫 𝐴 |
11 | 1 | canth 6508 | . . . . 5 ⊢ ¬ 𝑓:𝐴–onto→𝒫 𝐴 |
12 | f1ofo 6057 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→𝒫 𝐴 → 𝑓:𝐴–onto→𝒫 𝐴) | |
13 | 11, 12 | mto 187 | . . . 4 ⊢ ¬ 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
14 | 13 | nex 1722 | . . 3 ⊢ ¬ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴 |
15 | bren 7850 | . . 3 ⊢ (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝒫 𝐴) | |
16 | 14, 15 | mtbir 312 | . 2 ⊢ ¬ 𝐴 ≈ 𝒫 𝐴 |
17 | brsdom 7864 | . 2 ⊢ (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴)) | |
18 | 10, 16, 17 | mpbir2an 957 | 1 ⊢ 𝐴 ≺ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 –onto→wfo 5802 –1-1-onto→wf1o 5803 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: canth2g 7999 r1sdom 8520 alephsucpw2 8817 dfac13 8847 pwsdompw 8909 numthcor 9199 alephexp1 9280 pwcfsdom 9284 cfpwsdom 9285 gchhar 9380 gchac 9382 inawinalem 9390 tskcard 9482 gruina 9519 grothac 9531 rpnnen 14795 rexpen 14796 rucALT 14798 rectbntr0 22443 |
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