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 7567 | . . . 4 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) | |
2 | 1 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
3 | pweq 4558 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
4 | 3 | fveq2d 6677 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴)) |
5 | fveq2 6673 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴)) | |
6 | suceq 6259 | . . . . . . . 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 2840 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴))) |
9 | vex 3500 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
10 | 9 | rankpw 9275 | . . . . . 6 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥) |
11 | 8, 10 | vtoclg 3570 | . . . . 5 ⊢ (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴)) |
12 | 11 | eleq1d 2900 | . . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
13 | 12 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵)) |
14 | 2, 13 | bitr4d 284 | . 2 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
15 | fvprc 6666 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅) | |
16 | pwexb 7491 | . . . . . 6 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
17 | fvprc 6666 | . . . . . 6 ⊢ (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅) | |
18 | 16, 17 | sylnbi 332 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅) |
19 | 15, 18 | eqtr4d 2862 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴)) |
20 | 19 | eleq1d 2900 | . . 3 ⊢ (¬ 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) |
21 | 20 | adantr 483 | . 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 𝒫 cpw 4542 Lim wlim 6195 suc csuc 6196 ‘cfv 6358 rankcrnk 9195 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-reg 9059 ax-inf2 9107 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-r1 9196 df-rank 9197 |
This theorem is referenced by: rankxplim 9311 |
Copyright terms: Public domain | W3C validator |