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Mirrors > Home > MPE Home > Th. List > pw2en | Structured version Visualization version GIF version |
Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. This is Metamath 100 proof #52. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.) |
Ref | Expression |
---|---|
pw2en.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
pw2en | ⊢ 𝒫 𝐴 ≈ (2o ↑m 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw2en.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | pw2eng 9074 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝒫 𝐴 ≈ (2o ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3474 𝒫 cpw 4601 class class class wbr 5147 (class class class)co 7405 2oc2o 8456 ↑m cmap 8816 ≈ cen 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1o 8462 df-2o 8463 df-map 8818 df-en 8936 |
This theorem is referenced by: aleph1 10562 alephexp1 10570 pwcfsdom 10574 cfpwsdom 10575 hashpw 14392 rpnnen 16166 rexpen 16167 |
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