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

Theorem rankval4 9027
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 2934 . . . . . 6 𝑥𝐴
2 nfcv 2934 . . . . . . 7 𝑥𝑅1
3 nfiu1 4783 . . . . . . 7 𝑥 𝑥𝐴 suc (rank‘𝑥)
42, 3nffv 6456 . . . . . 6 𝑥(𝑅1 𝑥𝐴 suc (rank‘𝑥))
51, 4dfss2f 3812 . . . . 5 (𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝑅1 𝑥𝐴 suc (rank‘𝑥))))
6 vex 3401 . . . . . . 7 𝑥 ∈ V
76rankid 8993 . . . . . 6 𝑥 ∈ (𝑅1‘suc (rank‘𝑥))
8 ssiun2 4796 . . . . . . . 8 (𝑥𝐴 → suc (rank‘𝑥) ⊆ 𝑥𝐴 suc (rank‘𝑥))
9 rankon 8955 . . . . . . . . . 10 (rank‘𝑥) ∈ On
109onsuci 7316 . . . . . . . . 9 suc (rank‘𝑥) ∈ On
11 rankr1b.1 . . . . . . . . . 10 𝐴 ∈ V
1210rgenw 3106 . . . . . . . . . 10 𝑥𝐴 suc (rank‘𝑥) ∈ On
13 iunon 7719 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ∀𝑥𝐴 suc (rank‘𝑥) ∈ On) → 𝑥𝐴 suc (rank‘𝑥) ∈ On)
1411, 12, 13mp2an 682 . . . . . . . . 9 𝑥𝐴 suc (rank‘𝑥) ∈ On
15 r1ord3 8942 . . . . . . . . 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 682 . . . . . . . 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 3820 . . . . . 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 1843 . . . 4 𝐴 ⊆ (𝑅1 𝑥𝐴 suc (rank‘𝑥))
21 fvex 6459 . . . . 5 (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ∈ V
2221rankss 9009 . . . 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 8942 . . . . . . 7 (( 𝑥𝐴 suc (rank‘𝑥) ∈ On ∧ 𝑦 ∈ On) → ( 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)))
2514, 24mpan 680 . . . . . 6 (𝑦 ∈ On → ( 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦 → (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)))
2625ss2rabi 3905 . . . . 5 {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦} ⊆ {𝑦 ∈ On ∣ (𝑅1 𝑥𝐴 suc (rank‘𝑥)) ⊆ (𝑅1𝑦)}
27 intss 4731 . . . . 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 8978 . . . . 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 4730 . . . . . 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 2787 . . . 4 𝑥𝐴 suc (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥𝐴 suc (rank‘𝑥) ⊆ 𝑦}
3428, 30, 333sstr4i 3863 . . 3 (rank‘(𝑅1 𝑥𝐴 suc (rank‘𝑥))) ⊆ 𝑥𝐴 suc (rank‘𝑥)
3523, 34sstri 3830 . 2 (rank‘𝐴) ⊆ 𝑥𝐴 suc (rank‘𝑥)
36 iunss 4794 . . 3 ( 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴) ↔ ∀𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴))
3711rankel 8999 . . . 4 (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))
38 rankon 8955 . . . . 5 (rank‘𝐴) ∈ On
399, 38onsucssi 7319 . . . 4 ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ⊆ (rank‘𝐴))
4037, 39sylib 210 . . 3 (𝑥𝐴 → suc (rank‘𝑥) ⊆ (rank‘𝐴))
4136, 40mprgbir 3109 . 2 𝑥𝐴 suc (rank‘𝑥) ⊆ (rank‘𝐴)
4235, 41eqssi 3837 1 (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  wral 3090  {crab 3094  Vcvv 3398  wss 3792   cint 4710   ciun 4753  Oncon0 5976  suc csuc 5978  cfv 6135  𝑅1cr1 8922  rankcrnk 8923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-reg 8786  ax-inf2 8835
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-r1 8924  df-rank 8925
This theorem is referenced by:  rankbnd  9028  rankc1  9030
  Copyright terms: Public domain W3C validator