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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 8518 | . . . . 5 ⊢ 1o ∈ On | |
| 2 | pwdjuen 10222 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1o ∈ On) → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o)) | |
| 3 | 1, 2 | mpan2 691 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o)) |
| 4 | pwexg 5378 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
| 5 | 1oex 8516 | . . . . . 6 ⊢ 1o ∈ V | |
| 6 | 5 | pwex 5380 | . . . . 5 ⊢ 𝒫 1o ∈ V |
| 7 | xpcomeng 9104 | . . . . 5 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 1o ∈ V) → (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) | |
| 8 | 4, 6, 7 | sylancl 586 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) |
| 9 | entr 9046 | . . . 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 584 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 1o × 𝒫 𝐴)) |
| 11 | pwpw0 4813 | . . . . . 6 ⊢ 𝒫 {∅} = {∅, {∅}} | |
| 12 | df1o2 8513 | . . . . . . 7 ⊢ 1o = {∅} | |
| 13 | 12 | pweqi 4616 | . . . . . 6 ⊢ 𝒫 1o = 𝒫 {∅} |
| 14 | df2o2 8515 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
| 15 | 11, 13, 14 | 3eqtr4i 2775 | . . . . 5 ⊢ 𝒫 1o = 2o |
| 16 | 15 | xpeq1i 5711 | . . . 4 ⊢ (𝒫 1o × 𝒫 𝐴) = (2o × 𝒫 𝐴) |
| 17 | xp2dju 10217 | . . . 4 ⊢ (2o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) | |
| 18 | 16, 17 | eqtri 2765 | . . 3 ⊢ (𝒫 1o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) |
| 19 | 10, 18 | breqtrdi 5184 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴)) |
| 20 | 19 | ensymd 9045 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 (𝐴 ⊔ 1o)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {cpr 4628 class class class wbr 5143 × cxp 5683 Oncon0 6384 1oc1o 8499 2oc2o 8500 ≈ cen 8982 ⊔ cdju 9938 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-dju 9941 |
| This theorem is referenced by: pwdjuidm 10232 djulepw 10233 pwsdompw 10243 gchdjuidm 10708 gchpwdom 10710 |
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