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Mirrors > Home > MPE Home > Th. List > pw2eng | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
pw2eng | ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 4776 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | ovex 6577 | . . . 4 ⊢ ({∅, {∅}} ↑𝑚 𝐴) ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({∅, {∅}} ↑𝑚 𝐴) ∈ V) |
4 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
5 | 0ex 4718 | . . . . 5 ⊢ ∅ ∈ V | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
7 | p0ex 4779 | . . . . 5 ⊢ {∅} ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V) |
9 | 0nep0 4762 | . . . . 5 ⊢ ∅ ≠ {∅} | |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ≠ {∅}) |
11 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) | |
12 | 4, 6, 8, 10, 11 | pw2f1o 7950 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))):𝒫 𝐴–1-1-onto→({∅, {∅}} ↑𝑚 𝐴)) |
13 | f1oen2g 7858 | . . 3 ⊢ ((𝒫 𝐴 ∈ V ∧ ({∅, {∅}} ↑𝑚 𝐴) ∈ V ∧ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))):𝒫 𝐴–1-1-onto→({∅, {∅}} ↑𝑚 𝐴)) → 𝒫 𝐴 ≈ ({∅, {∅}} ↑𝑚 𝐴)) | |
14 | 1, 3, 12, 13 | syl3anc 1318 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ ({∅, {∅}} ↑𝑚 𝐴)) |
15 | df2o2 7461 | . . 3 ⊢ 2𝑜 = {∅, {∅}} | |
16 | 15 | oveq1i 6559 | . 2 ⊢ (2𝑜 ↑𝑚 𝐴) = ({∅, {∅}} ↑𝑚 𝐴) |
17 | 14, 16 | syl6breqr 4625 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 {csn 4125 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 –1-1-onto→wf1o 5803 (class class class)co 6549 2𝑜c2o 7441 ↑𝑚 cmap 7744 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1o 7447 df-2o 7448 df-map 7746 df-en 7842 |
This theorem is referenced by: pw2en 7952 pwen 8018 mappwen 8818 pwcdaen 8890 hauspwdom 21114 enrelmap 37311 |
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