![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7858 | . . . 4 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) | |
2 | 1 | adantl 480 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
3 | pweq 4620 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
4 | 3 | fveq2d 6904 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴)) |
5 | fveq2 6900 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴)) | |
6 | suceq 6441 | . . . . . . . 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 2741 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴))) |
9 | vex 3465 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | 9 | rankpw 9882 | . . . . . 6 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥) |
11 | 8, 10 | vtoclg 3533 | . . . . 5 ⊢ (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴)) |
12 | 11 | eleq1d 2810 | . . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
13 | 12 | adantr 479 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
14 | 2, 13 | bitr4d 281 | . 2 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
15 | fvprc 6892 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅) | |
16 | pwexb 7773 | . . . . . 6 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
17 | fvprc 6892 | . . . . . 6 ⊢ (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅) | |
18 | 16, 17 | sylnbi 329 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅) |
19 | 15, 18 | eqtr4d 2768 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴)) |
20 | 19 | eleq1d 2810 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
21 | 20 | adantr 479 | . 2 ⊢ ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
22 | 14, 21 | pm2.61ian 810 | 1 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4324 𝒫 cpw 4606 Lim wlim 6376 suc csuc 6377 ‘cfv 6553 rankcrnk 9802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-reg 9631 ax-inf2 9680 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-r1 9803 df-rank 9804 |
This theorem is referenced by: rankxplim 9918 |
Copyright terms: Public domain | W3C validator |