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Mirrors > Home > ILE Home > Th. List > canth | Unicode version |
Description: No set is equinumerous to its power set (Cantor's theorem), i.e., no function can map onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1493 if you want the form .) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.) |
Ref | Expression |
---|---|
canth.1 |
Ref | Expression |
---|---|
canth |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | canth.1 | . . . 4 | |
2 | ssrab2 3232 | . . . 4 | |
3 | 1, 2 | elpwi2 4144 | . . 3 |
4 | forn 5423 | . . 3 | |
5 | 3, 4 | eleqtrrid 2260 | . 2 |
6 | pm5.19 701 | . . . . . 6 | |
7 | eleq2 2234 | . . . . . . 7 | |
8 | id 19 | . . . . . . . . . 10 | |
9 | fveq2 5496 | . . . . . . . . . 10 | |
10 | 8, 9 | eleq12d 2241 | . . . . . . . . 9 |
11 | 10 | notbid 662 | . . . . . . . 8 |
12 | 11 | elrab3 2887 | . . . . . . 7 |
13 | 7, 12 | sylan9bbr 460 | . . . . . 6 |
14 | 6, 13 | mto 657 | . . . . 5 |
15 | 14 | imnani 686 | . . . 4 |
16 | 15 | nrex 2562 | . . 3 |
17 | fofn 5422 | . . . 4 | |
18 | fvelrnb 5544 | . . . 4 | |
19 | 17, 18 | syl 14 | . . 3 |
20 | 16, 19 | mtbiri 670 | . 2 |
21 | 5, 20 | pm2.65i 634 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wceq 1348 wcel 2141 wrex 2449 crab 2452 cvv 2730 cpw 3566 crn 4612 wfn 5193 wfo 5196 cfv 5198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 |
This theorem is referenced by: (None) |
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