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Mirrors > Home > MPE Home > Th. List > ranklim | Structured version Visualization version GIF version |
Description: The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.) |
Ref | Expression |
---|---|
ranklim | ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsuc 7577 | . . . 4 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) | |
2 | 1 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
3 | pweq 4501 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
4 | 3 | fveq2d 6672 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴)) |
5 | fveq2 6668 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴)) | |
6 | suceq 6231 | . . . . . . . 8 ⊢ ((rank‘𝑥) = (rank‘𝐴) → suc (rank‘𝑥) = suc (rank‘𝐴)) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴)) |
8 | 4, 7 | eqeq12d 2754 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴))) |
9 | vex 3401 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | 9 | rankpw 9338 | . . . . . 6 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥) |
11 | 8, 10 | vtoclg 3470 | . . . . 5 ⊢ (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴)) |
12 | 11 | eleq1d 2817 | . . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
13 | 12 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
14 | 2, 13 | bitr4d 285 | . 2 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
15 | fvprc 6660 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅) | |
16 | pwexb 7501 | . . . . . 6 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
17 | fvprc 6660 | . . . . . 6 ⊢ (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅) | |
18 | 16, 17 | sylnbi 333 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅) |
19 | 15, 18 | eqtr4d 2776 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴)) |
20 | 19 | eleq1d 2817 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
21 | 20 | adantr 484 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
22 | 14, 21 | pm2.61ian 812 | 1 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 Vcvv 3397 ∅c0 4209 𝒫 cpw 4485 Lim wlim 6167 suc csuc 6168 ‘cfv 6333 rankcrnk 9258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-reg 9122 ax-inf2 9170 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-r1 9259 df-rank 9260 |
This theorem is referenced by: rankxplim 9374 |
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