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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 1480 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 3213 | . . . 4 | |
3 | 1, 2 | elpwi2 4120 | . . 3 |
4 | forn 5396 | . . 3 | |
5 | 3, 4 | eleqtrrid 2247 | . 2 |
6 | pm5.19 696 | . . . . . 6 | |
7 | eleq2 2221 | . . . . . . 7 | |
8 | id 19 | . . . . . . . . . 10 | |
9 | fveq2 5469 | . . . . . . . . . 10 | |
10 | 8, 9 | eleq12d 2228 | . . . . . . . . 9 |
11 | 10 | notbid 657 | . . . . . . . 8 |
12 | 11 | elrab3 2869 | . . . . . . 7 |
13 | 7, 12 | sylan9bbr 459 | . . . . . 6 |
14 | 6, 13 | mto 652 | . . . . 5 |
15 | 14 | imnani 681 | . . . 4 |
16 | 15 | nrex 2549 | . . 3 |
17 | fofn 5395 | . . . 4 | |
18 | fvelrnb 5517 | . . . 4 | |
19 | 17, 18 | syl 14 | . . 3 |
20 | 16, 19 | mtbiri 665 | . 2 |
21 | 5, 20 | pm2.65i 629 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wceq 1335 wcel 2128 wrex 2436 crab 2439 cvv 2712 cpw 3543 crn 4588 wfn 5166 wfo 5169 cfv 5171 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fo 5177 df-fv 5179 |
This theorem is referenced by: (None) |
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