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Mirrors > Home > MPE Home > Th. List > pwdju1 | Structured version Visualization version GIF version |
Description: The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
pwdju1 | ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 (𝐴 ⊔ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 8109 | . . . . 5 ⊢ 1o ∈ On | |
2 | pwdjuen 9607 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1o ∈ On) → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o)) | |
3 | 1, 2 | mpan2 689 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o)) |
4 | pwexg 5279 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
5 | 1oex 8110 | . . . . . 6 ⊢ 1o ∈ V | |
6 | 5 | pwex 5281 | . . . . 5 ⊢ 𝒫 1o ∈ V |
7 | xpcomeng 8609 | . . . . 5 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 1o ∈ V) → (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) | |
8 | 4, 6, 7 | sylancl 588 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) |
9 | entr 8561 | . . . 4 ⊢ ((𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o) ∧ (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 1o × 𝒫 𝐴)) | |
10 | 3, 8, 9 | syl2anc 586 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 1o × 𝒫 𝐴)) |
11 | pwpw0 4746 | . . . . . 6 ⊢ 𝒫 {∅} = {∅, {∅}} | |
12 | df1o2 8116 | . . . . . . 7 ⊢ 1o = {∅} | |
13 | 12 | pweqi 4557 | . . . . . 6 ⊢ 𝒫 1o = 𝒫 {∅} |
14 | df2o2 8118 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
15 | 11, 13, 14 | 3eqtr4i 2854 | . . . . 5 ⊢ 𝒫 1o = 2o |
16 | 15 | xpeq1i 5581 | . . . 4 ⊢ (𝒫 1o × 𝒫 𝐴) = (2o × 𝒫 𝐴) |
17 | xp2dju 9602 | . . . 4 ⊢ (2o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) | |
18 | 16, 17 | eqtri 2844 | . . 3 ⊢ (𝒫 1o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) |
19 | 10, 18 | breqtrdi 5107 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴)) |
20 | 19 | ensymd 8560 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 (𝐴 ⊔ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 𝒫 cpw 4539 {csn 4567 {cpr 4569 class class class wbr 5066 × cxp 5553 Oncon0 6191 1oc1o 8095 2oc2o 8096 ≈ cen 8506 ⊔ cdju 9327 |
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 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-1o 8102 df-2o 8103 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-dju 9330 |
This theorem is referenced by: pwdjuidm 9617 djulepw 9618 pwsdompw 9626 gchdjuidm 10090 gchpwdom 10092 |
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