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Theorem ranklim 9882
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.)
Assertion
Ref Expression
ranklim (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))

Proof of Theorem ranklim
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 limsuc 7870 . . . 4 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
21adantl 481 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
3 pweq 4619 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43fveq2d 6911 . . . . . . 7 (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
5 fveq2 6907 . . . . . . . 8 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
6 suceq 6452 . . . . . . . 8 ((rank‘𝑥) = (rank‘𝐴) → suc (rank‘𝑥) = suc (rank‘𝐴))
75, 6syl 17 . . . . . . 7 (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴))
84, 7eqeq12d 2751 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
9 vex 3482 . . . . . . 7 𝑥 ∈ V
109rankpw 9881 . . . . . 6 (rank‘𝒫 𝑥) = suc (rank‘𝑥)
118, 10vtoclg 3554 . . . . 5 (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1211eleq1d 2824 . . . 4 (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1312adantr 480 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
142, 13bitr4d 282 . 2 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
15 fvprc 6899 . . . . 5 𝐴 ∈ V → (rank‘𝐴) = ∅)
16 pwexb 7785 . . . . . 6 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
17 fvprc 6899 . . . . . 6 (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1816, 17sylnbi 330 . . . . 5 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1915, 18eqtr4d 2778 . . . 4 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴))
2019eleq1d 2824 . . 3 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2120adantr 480 . 2 ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2214, 21pm2.61ian 812 1 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  𝒫 cpw 4605  Lim wlim 6387  suc csuc 6388  cfv 6563  rankcrnk 9801
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
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-reu 3379  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-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  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-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803
This theorem is referenced by:  rankxplim  9917
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