MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankval4 Structured version   Visualization version   GIF version

Theorem rankval4 9907
Description: The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.)
Hypothesis
Ref Expression
rankr1b.1 𝐴 ∈ V
Assertion
Ref Expression
rankval4 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankval4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2905 . . . . . 6 𝑥𝐴
2 nfcv 2905 . . . . . . 7 𝑥𝑅1
3 nfiu1 5027 . . . . . . 7 𝑥 𝑥𝐴 suc (rank‘𝑥)
42, 3nffv 6916 . . . . . 6 𝑥(𝑅1 𝑥𝐴 suc (rank‘𝑥))
51, 4dfssf 3974 . . . . 5 (𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝑅1 𝑥𝐴 suc (rank‘𝑥))))
6 vex 3484 . . . . . . 7 𝑥 ∈ V
76rankid 9873 . . . . . 6 𝑥 ∈ (𝑅1‘suc (rank‘𝑥))
8 ssiun2 5047 . . . . . . . 8 (𝑥𝐴 → suc (rank‘𝑥) ⊆ 𝑥𝐴 suc (rank‘𝑥))
9 rankon 9835 . . . . . . . . . 10 (rank‘𝑥) ∈ On
109onsuci 7859 . . . . . . . . 9 suc (rank‘𝑥) ∈ On
11 rankr1b.1 . . . . . . . . . 10 𝐴 ∈ V
1210rgenw 3065 . . . . . . . . . 10 𝑥𝐴 suc (rank‘𝑥) ∈ On
13 iunon 8379 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ∀𝑥𝐴 suc (rank‘𝑥) ∈ On) → 𝑥𝐴 suc (rank‘𝑥) ∈ On)
1411, 12, 13mp2an 692 . . . . . . . . 9 𝑥𝐴 suc (rank‘𝑥) ∈ On
15 r1ord3 9822 . . . . . . . . 9 ((suc (rank‘𝑥) ∈ On ∧ 𝑥𝐴 suc (rank‘𝑥) ∈ On) → (suc (rank‘𝑥) ⊆ 𝑥𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥))))
1610, 14, 15mp2an 692 . . . . . . . 8 (suc (rank‘𝑥) ⊆ 𝑥𝐴 suc (rank‘𝑥) → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)))
178, 16syl 17 . . . . . . 7 (𝑥𝐴 → (𝑅1‘suc (rank‘𝑥)) ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)))
1817sseld 3982 . . . . . 6 (𝑥𝐴 → (𝑥 ∈ (𝑅1‘suc (rank‘𝑥)) → 𝑥 ∈ (𝑅1 𝑥𝐴 suc (rank‘𝑥))))
197, 18mpi 20 . . . . 5 (𝑥𝐴𝑥 ∈ (𝑅1 𝑥𝐴 suc (rank‘𝑥)))
205, 19mpgbir 1799 . . . 4 𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥))
21 fvex 6919 . . . . 5 (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ∈ V
2221rankss 9889 . . . 4 (𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) → (rank‘𝐴) ⊆ (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))))
2320, 22ax-mp 5 . . 3 (rank‘𝐴) ⊆ (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥)))
24 r1ord3 9822 . . . . . . 7 (( 𝑥𝐴 suc (rank‘𝑥) ∈ On ∧ 𝑦 ∈ On) → ( 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)))
2514, 24mpan 690 . . . . . 6 (𝑦 ∈ On → ( 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)))
2625ss2rabi 4077 . . . . 5 {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)}
27 intss 4969 . . . . 5 ({𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)} → {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)} ⊆ {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦})
2826, 27ax-mp 5 . . . 4 {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)} ⊆ {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦}
29 rankval2 9858 . . . . 5 ((𝑅1 𝑥𝐴 suc (rank‘𝑥)) ∈ V → (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))) = {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)})
3021, 29ax-mp 5 . . . 4 (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))) = {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)}
31 intmin 4968 . . . . . 6 ( 𝑥𝐴 suc (rank‘𝑥) ∈ On → {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} = 𝑥𝐴 suc (rank‘𝑥))
3214, 31ax-mp 5 . . . . 5 {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} = 𝑥𝐴 suc (rank‘𝑥)
3332eqcomi 2746 . . . 4 𝑥𝐴 suc (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦}
3428, 30, 333sstr4i 4035 . . 3 (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))) ⊆ 𝑥𝐴 suc (rank‘𝑥)
3523, 34sstri 3993 . 2 (rank‘𝐴) ⊆ 𝑥𝐴 suc (rank‘𝑥)
36 iunss 5045 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
3711rankel 9879 . . . 4 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
38 rankon 9835 . . . . 5 (rank‘𝐴) ∈ On
399, 38onsucssi 7862 . . . 4 ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘𝐴))
4037, 39sylib 218 . . 3 (𝑥𝐴 → suc (rank‘𝑥) ⊆ (rank‘𝐴))
4136, 40mprgbir 3068 . 2 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴)
4235, 41eqssi 4000 1 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951   cint 4946   ciun 4991  Oncon0 6384  suc csuc 6386  cfv 6561  𝑅1cr1 9802  rankcrnk 9803
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  rankbnd  9908  rankc1  9910  scottrankd  44267
  Copyright terms: Public domain W3C validator