Theorem ranklim 9743

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.

Step	Hyp	Ref	Expression
1		limsuc 7785	. . . 4 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
2	1	adantl 481	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
3		pweq 4563	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
4	3	fveq2d 6832	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
5		fveq2 6828	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
6		suceq 6380	. . . . . . . 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 2747	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
9		vex 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	rankpw 9742	. . . . . 6 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥)
11	8, 10	vtoclg 3507	. . . . 5 ⊢ (𝐴 ∈ V → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
12	11	eleq1d 2816	. . . 4 ⊢ (𝐴 ∈ V → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
13	12	adantr 480	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝒫 𝐴) ∈ 𝐵 ↔ suc (rank‘𝐴) ∈ 𝐵))
14	2, 13	bitr4d 282	. 2 ⊢ ((𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
15		fvprc 6820	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
16		pwexb 7705	. . . . . 6 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
17		fvprc 6820	. . . . . 6 ⊢ (¬ 𝒫 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
18	16, 17	sylnbi 330	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (rank‘𝒫 𝐴) = ∅)
19	15, 18	eqtr4d 2769	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = (rank‘𝒫 𝐴))
20	19	eleq1d 2816	. . 3 ⊢ (¬ 𝐴 ∈ V → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
21	20	adantr 480	. 2 ⊢ ((¬ 𝐴 ∈ V ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))
22	14, 21	pm2.61ian 811	1 ⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4282  𝒫 cpw 4549  Lim wlim 6313  suc csuc 6314  ‘cfv 6487  rankcrnk 9662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-reg 9484  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-r1 9663  df-rank 9664

This theorem is referenced by:  rankxplim  9778