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Mirrors > Home > MPE Home > Th. List > Mathboxes > rankpwg | Structured version Visualization version GIF version |
Description: The rank of a power set. Closed form of rankpw 9879. (Contributed by Scott Fenton, 16-Jul-2015.) |
Ref | Expression |
---|---|
rankpwg | ⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4612 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
2 | 1 | fveq2d 6908 | . . 3 ⊢ (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴)) |
3 | fveq2 6904 | . . . 4 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴)) | |
4 | suceq 6448 | . . . 4 ⊢ ((rank‘𝑥) = (rank‘𝐴) → suc (rank‘𝑥) = suc (rank‘𝐴)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴)) |
6 | 2, 5 | eqeq12d 2752 | . 2 ⊢ (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴))) |
7 | vex 3483 | . . 3 ⊢ 𝑥 ∈ V | |
8 | 7 | rankpw 9879 | . 2 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥) |
9 | 6, 8 | vtoclg 3553 | 1 ⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 𝒫 cpw 4598 suc csuc 6384 ‘cfv 6559 rankcrnk 9799 |
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 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-reg 9628 ax-inf2 9677 |
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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-om 7884 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-r1 9800 df-rank 9801 |
This theorem is referenced by: hfpw 36164 |
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