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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 5338 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | ovexd 7397 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({∅, {∅}} ↑m 𝐴) ∈ V) | |
3 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
4 | 0ex 5269 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
6 | p0ex 5344 | . . . . 5 ⊢ {∅} ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V) |
8 | 0nep0 5318 | . . . . 5 ⊢ ∅ ≠ {∅} | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ≠ {∅}) |
10 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) | |
11 | 3, 5, 7, 9, 10 | pw2f1o 9028 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))):𝒫 𝐴–1-1-onto→({∅, {∅}} ↑m 𝐴)) |
12 | f1oen2g 8915 | . . 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)) | |
13 | 1, 2, 11, 12 | syl3anc 1372 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ ({∅, {∅}} ↑m 𝐴)) |
14 | df2o2 8426 | . . 3 ⊢ 2o = {∅, {∅}} | |
15 | 14 | oveq1i 7372 | . 2 ⊢ (2o ↑m 𝐴) = ({∅, {∅}} ↑m 𝐴) |
16 | 13, 15 | breqtrrdi 5152 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 ∅c0 4287 ifcif 4491 𝒫 cpw 4565 {csn 4591 {cpr 4593 class class class wbr 5110 ↦ cmpt 5193 –1-1-onto→wf1o 6500 (class class class)co 7362 2oc2o 8411 ↑m cmap 8772 ≈ cen 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1o 8417 df-2o 8418 df-map 8774 df-en 8891 |
This theorem is referenced by: pw2en 9030 pwen 9101 mappwen 10055 pwdjuen 10124 ackbij1lem5 10167 hauspwdom 22868 enrelmap 42343 |
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