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Theorem pw2eng 8308
Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal 2𝑜. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.)
Assertion
Ref Expression
pw2eng (𝐴𝑉 → 𝒫 𝐴 ≈ (2𝑜𝑚 𝐴))

Proof of Theorem pw2eng
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 5048 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 ovexd 6912 . . 3 (𝐴𝑉 → ({∅, {∅}} ↑𝑚 𝐴) ∈ V)
3 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
4 0ex 4984 . . . . 5 ∅ ∈ V
54a1i 11 . . . 4 (𝐴𝑉 → ∅ ∈ V)
6 p0ex 5053 . . . . 5 {∅} ∈ V
76a1i 11 . . . 4 (𝐴𝑉 → {∅} ∈ V)
8 0nep0 5028 . . . . 5 ∅ ≠ {∅}
98a1i 11 . . . 4 (𝐴𝑉 → ∅ ≠ {∅})
10 eqid 2799 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, {∅}, ∅))) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, {∅}, ∅)))
113, 5, 7, 9, 10pw2f1o 8307 . . 3 (𝐴𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, {∅}, ∅))):𝒫 𝐴1-1-onto→({∅, {∅}} ↑𝑚 𝐴))
12 f1oen2g 8212 . . 3 ((𝒫 𝐴 ∈ V ∧ ({∅, {∅}} ↑𝑚 𝐴) ∈ V ∧ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, {∅}, ∅))):𝒫 𝐴1-1-onto→({∅, {∅}} ↑𝑚 𝐴)) → 𝒫 𝐴 ≈ ({∅, {∅}} ↑𝑚 𝐴))
131, 2, 11, 12syl3anc 1491 . 2 (𝐴𝑉 → 𝒫 𝐴 ≈ ({∅, {∅}} ↑𝑚 𝐴))
14 df2o2 7814 . . 3 2𝑜 = {∅, {∅}}
1514oveq1i 6888 . 2 (2𝑜𝑚 𝐴) = ({∅, {∅}} ↑𝑚 𝐴)
1613, 15syl6breqr 4885 1 (𝐴𝑉 → 𝒫 𝐴 ≈ (2𝑜𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wne 2971  Vcvv 3385  c0 4115  ifcif 4277  𝒫 cpw 4349  {csn 4368  {cpr 4370   class class class wbr 4843  cmpt 4922  1-1-ontowf1o 6100  (class class class)co 6878  2𝑜c2o 7793  𝑚 cmap 8095  cen 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1o 7799  df-2o 7800  df-map 8097  df-en 8196
This theorem is referenced by:  pw2en  8309  pwen  8375  mappwen  9221  pwcdaen  9295  ackbij1lem5  9334  hauspwdom  21633  enrelmap  39069
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