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Theorem rankprb 8674
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 8632 . . . 4 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
2 snwf 8632 . . . 4 (𝐵 (𝑅1 “ On) → {𝐵} ∈ (𝑅1 “ On))
3 rankunb 8673 . . . 4 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐵})))
41, 2, 3syl2an 494 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐵})))
5 ranksnb 8650 . . . 4 (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))
6 ranksnb 8650 . . . 4 (𝐵 (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
7 uneq12 3746 . . . 4 (((rank‘{𝐴}) = suc (rank‘𝐴) ∧ (rank‘{𝐵}) = suc (rank‘𝐵)) → ((rank‘{𝐴}) ∪ (rank‘{𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
85, 6, 7syl2an 494 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
94, 8eqtrd 2655 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
10 df-pr 4158 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
1110fveq2i 6161 . 2 (rank‘{𝐴, 𝐵}) = (rank‘({𝐴} ∪ {𝐵}))
12 rankon 8618 . . . 4 (rank‘𝐴) ∈ On
1312onordi 5801 . . 3 Ord (rank‘𝐴)
14 rankon 8618 . . . 4 (rank‘𝐵) ∈ On
1514onordi 5801 . . 3 Ord (rank‘𝐵)
16 ordsucun 6987 . . 3 ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘𝐵)) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
1713, 15, 16mp2an 707 . 2 suc ((rank‘𝐴) ∪ (rank‘𝐵)) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵))
189, 11, 173eqtr4g 2680 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 384   = wceq 1480  wcel 1987  cun 3558  {csn 4155  {cpr 4157   cuni 4409  cima 5087  Ord word 5691  Oncon0 5692  suc csuc 5694  cfv 5857  𝑅1cr1 8585  rankcrnk 8586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-r1 8587  df-rank 8588
This theorem is referenced by:  rankopb  8675  rankpr  8680  r1limwun  9518  rankaltopb  31781
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