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Theorem pw2eng 8615
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 5270 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 ovexd 7183 . . 3 (𝐴𝑉 → ({∅, {∅}} ↑m 𝐴) ∈ V)
3 id 22 . . . 4 (𝐴𝑉𝐴𝑉)
4 0ex 5202 . . . . 5 ∅ ∈ V
54a1i 11 . . . 4 (𝐴𝑉 → ∅ ∈ V)
6 p0ex 5275 . . . . 5 {∅} ∈ V
76a1i 11 . . . 4 (𝐴𝑉 → {∅} ∈ V)
8 0nep0 5249 . . . . 5 ∅ ≠ {∅}
98a1i 11 . . . 4 (𝐴𝑉 → ∅ ≠ {∅})
10 eqid 2819 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, {∅}, ∅))) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, {∅}, ∅)))
113, 5, 7, 9, 10pw2f1o 8614 . . 3 (𝐴𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, {∅}, ∅))):𝒫 𝐴1-1-onto→({∅, {∅}} ↑m 𝐴))
12 f1oen2g 8518 . . 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 1366 . 2 (𝐴𝑉 → 𝒫 𝐴 ≈ ({∅, {∅}} ↑m 𝐴))
14 df2o2 8110 . . 3 2o = {∅, {∅}}
1514oveq1i 7158 . 2 (2om 𝐴) = ({∅, {∅}} ↑m 𝐴)
1613, 15breqtrrdi 5099 1 (𝐴𝑉 → 𝒫 𝐴 ≈ (2om 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wne 3014  Vcvv 3493  c0 4289  ifcif 4465  𝒫 cpw 4537  {csn 4559  {cpr 4561   class class class wbr 5057  cmpt 5137  1-1-ontowf1o 6347  (class class class)co 7148  2oc2o 8088  m cmap 8398  cen 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1o 8094  df-2o 8095  df-map 8400  df-en 8502
This theorem is referenced by:  pw2en  8616  pwen  8682  mappwen  9530  pwdjuen  9599  ackbij1lem5  9638  hauspwdom  22101  enrelmap  40333
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