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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 8419 | . 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 2050 Vcvv 3415 𝒫 cpw 4422 class class class wbr 4929 (class class class)co 6976 2oc2o 7899 ↑𝑚 cmap 8206 ≈ cen 8303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-1o 7905 df-2o 7906 df-map 8208 df-en 8307 |
This theorem is referenced by: aleph1 9791 alephexp1 9799 pwcfsdom 9803 cfpwsdom 9804 hashpw 13610 rpnnen 15440 rexpen 15441 |
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