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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 8517 | . . . . 5 ⊢ 1o ∈ On | |
2 | pwdjuen 10220 | . . . . 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 5384 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
5 | 1oex 8515 | . . . . . 6 ⊢ 1o ∈ V | |
6 | 5 | pwex 5386 | . . . . 5 ⊢ 𝒫 1o ∈ V |
7 | xpcomeng 9103 | . . . . 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 9045 | . . . 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 4818 | . . . . . 6 ⊢ 𝒫 {∅} = {∅, {∅}} | |
12 | df1o2 8512 | . . . . . . 7 ⊢ 1o = {∅} | |
13 | 12 | pweqi 4621 | . . . . . 6 ⊢ 𝒫 1o = 𝒫 {∅} |
14 | df2o2 8514 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
15 | 11, 13, 14 | 3eqtr4i 2773 | . . . . 5 ⊢ 𝒫 1o = 2o |
16 | 15 | xpeq1i 5715 | . . . 4 ⊢ (𝒫 1o × 𝒫 𝐴) = (2o × 𝒫 𝐴) |
17 | xp2dju 10215 | . . . 4 ⊢ (2o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) | |
18 | 16, 17 | eqtri 2763 | . . 3 ⊢ (𝒫 1o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) |
19 | 10, 18 | breqtrdi 5189 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴)) |
20 | 19 | ensymd 9044 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 (𝐴 ⊔ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 𝒫 cpw 4605 {csn 4631 {cpr 4633 class class class wbr 5148 × cxp 5687 Oncon0 6386 1oc1o 8498 2oc2o 8499 ≈ cen 8981 ⊔ cdju 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-dju 9939 |
This theorem is referenced by: pwdjuidm 10230 djulepw 10231 pwsdompw 10241 gchdjuidm 10706 gchpwdom 10708 |
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