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Theorem ranklim 9883
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 7858 . . . 4 (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
21adantl 480 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
3 pweq 4620 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43fveq2d 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))
75, 6syl 17 . . . . . . 7 (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴))
84, 7eqeq12d 2741 . . . . . 6 (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
9 vex 3465 . . . . . . 7 𝑥 ∈ V
109rankpw 9882 . . . . . 6 (rank‘𝒫 𝑥) = suc (rank‘𝑥)
118, 10vtoclg 3533 . . . . 5 (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
1211eleq1d 2810 . . . 4 (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
1312adantr 479 . . 3 ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
142, 13bitr4d 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) = ∅)
1816, 17sylnbi 329 . . . . 5 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
1915, 18eqtr4d 2768 . . . 4 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴))
2019eleq1d 2810 . . 3 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2120adantr 479 . 2 ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
2214, 21pm2.61ian 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
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