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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 5265 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
2 | ovexd 7177 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({∅, {∅}} ↑m 𝐴) ∈ V) | |
3 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
4 | 0ex 5197 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V) |
6 | p0ex 5271 | . . . . 5 ⊢ {∅} ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V) |
8 | 0nep0 5244 | . . . . 5 ⊢ ∅ ≠ {∅} | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ≠ {∅}) |
10 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) | |
11 | 3, 5, 7, 9, 10 | pw2f1o 8608 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))):𝒫 𝐴–1-1-onto→({∅, {∅}} ↑m 𝐴)) |
12 | f1oen2g 8512 | . . 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 1367 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ ({∅, {∅}} ↑m 𝐴)) |
14 | df2o2 8104 | . . 3 ⊢ 2o = {∅, {∅}} | |
15 | 14 | oveq1i 7152 | . 2 ⊢ (2o ↑m 𝐴) = ({∅, {∅}} ↑m 𝐴) |
16 | 13, 15 | breqtrrdi 5094 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3016 Vcvv 3486 ∅c0 4279 ifcif 4453 𝒫 cpw 4525 {csn 4553 {cpr 4555 class class class wbr 5052 ↦ cmpt 5132 –1-1-onto→wf1o 6340 (class class class)co 7142 2oc2o 8082 ↑m cmap 8392 ≈ cen 8492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-1o 8088 df-2o 8089 df-map 8394 df-en 8496 |
This theorem is referenced by: pw2en 8610 pwen 8676 mappwen 9524 pwdjuen 9593 ackbij1lem5 9632 hauspwdom 22092 enrelmap 40433 |
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