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

Theorem ranksnb 9274
 Description: The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
ranksnb (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))

Proof of Theorem ranksnb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6651 . . . . . 6 (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
21eleq1d 2835 . . . . 5 (𝑦 = 𝐴 → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
32ralsng 4563 . . . 4 (𝐴 (𝑅1 “ On) → (∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝐴) ∈ 𝑥))
43rabbidv 3390 . . 3 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
54inteqd 4836 . 2 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥} = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
6 snwf 9256 . . 3 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
7 rankval3b 9273 . . 3 ({𝐴} ∈ (𝑅1 “ On) → (rank‘{𝐴}) = {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥})
86, 7syl 17 . 2 (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = {𝑥 ∈ On ∣ ∀𝑦 ∈ {𝐴} (rank‘𝑦) ∈ 𝑥})
9 rankon 9242 . . 3 (rank‘𝐴) ∈ On
10 onsucmin 7528 . . 3 ((rank‘𝐴) ∈ On → suc (rank‘𝐴) = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
119, 10mp1i 13 . 2 (𝐴 (𝑅1 “ On) → suc (rank‘𝐴) = {𝑥 ∈ On ∣ (rank‘𝐴) ∈ 𝑥})
125, 8, 113eqtr4d 2804 1 (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  ∀wral 3068  {crab 3072  {csn 4515  ∪ cuni 4791  ∩ cint 4831   “ cima 5520  Oncon0 6162  suc csuc 6164  ‘cfv 6328  𝑅1cr1 9209  rankcrnk 9210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7573  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-r1 9211  df-rank 9212 This theorem is referenced by:  rankprb  9298  ranksn  9301  rankcf  10222  rankaltopb  33815
 Copyright terms: Public domain W3C validator