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

Theorem rankprb 8998
 Description: The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
rankprb ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankprb
StepHypRef Expression
1 snwf 8956 . . . 4 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
2 snwf 8956 . . . 4 (𝐵 (𝑅1 “ On) → {𝐵} ∈ (𝑅1 “ On))
3 rankunb 8997 . . . 4 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐵})))
41, 2, 3syl2an 589 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐵})))
5 ranksnb 8974 . . . 4 (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))
6 ranksnb 8974 . . . 4 (𝐵 (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
7 uneq12 3991 . . . 4 (((rank‘{𝐴}) = suc (rank‘𝐴) ∧ (rank‘{𝐵}) = suc (rank‘𝐵)) → ((rank‘{𝐴}) ∪ (rank‘{𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
85, 6, 7syl2an 589 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
94, 8eqtrd 2861 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
10 df-pr 4402 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
1110fveq2i 6440 . 2 (rank‘{𝐴, 𝐵}) = (rank‘({𝐴} ∪ {𝐵}))
12 rankon 8942 . . . 4 (rank‘𝐴) ∈ On
1312onordi 6071 . . 3 Ord (rank‘𝐴)
14 rankon 8942 . . . 4 (rank‘𝐵) ∈ On
1514onordi 6071 . . 3 Ord (rank‘𝐵)
16 ordsucun 7291 . . 3 ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘𝐵)) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
1713, 15, 16mp2an 683 . 2 suc ((rank‘𝐴) ∪ (rank‘𝐵)) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵))
189, 11, 173eqtr4g 2886 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ∪ cun 3796  {csn 4399  {cpr 4401  ∪ cuni 4660   “ cima 5349  Ord word 5966  Oncon0 5967  suc csuc 5969  ‘cfv 6127  𝑅1cr1 8909  rankcrnk 8910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-r1 8911  df-rank 8912 This theorem is referenced by:  rankopb  8999  rankpr  9004  r1limwun  9880  rankaltopb  32620
 Copyright terms: Public domain W3C validator