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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 5378 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | ovexd 7455 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({∅, {∅}} ↑m 𝐴) ∈ V) | |
3 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
4 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
6 | p0ex 5384 | . . . . 5 ⊢ {∅} ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V) |
8 | 0nep0 5358 | . . . . 5 ⊢ ∅ ≠ {∅} | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ≠ {∅}) |
10 | eqid 2728 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) | |
11 | 3, 5, 7, 9, 10 | pw2f1o 9102 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))):𝒫 𝐴–1-1-onto→({∅, {∅}} ↑m 𝐴)) |
12 | f1oen2g 8989 | . . 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 1369 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ ({∅, {∅}} ↑m 𝐴)) |
14 | df2o2 8496 | . . 3 ⊢ 2o = {∅, {∅}} | |
15 | 14 | oveq1i 7430 | . 2 ⊢ (2o ↑m 𝐴) = ({∅, {∅}} ↑m 𝐴) |
16 | 13, 15 | breqtrrdi 5190 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∅c0 4323 ifcif 4529 𝒫 cpw 4603 {csn 4629 {cpr 4631 class class class wbr 5148 ↦ cmpt 5231 –1-1-onto→wf1o 6547 (class class class)co 7420 2oc2o 8481 ↑m cmap 8845 ≈ cen 8961 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1o 8487 df-2o 8488 df-map 8847 df-en 8965 |
This theorem is referenced by: pw2en 9104 pwen 9175 mappwen 10136 pwdjuen 10205 ackbij1lem5 10248 hauspwdom 23418 enrelmap 43427 |
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