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Theorem rankprb 9271
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 9229 . . . 4 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
2 snwf 9229 . . . 4 (𝐵 (𝑅1 “ On) → {𝐵} ∈ (𝑅1 “ On))
3 rankunb 9270 . . . 4 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐵})))
41, 2, 3syl2an 598 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐵})))
5 ranksnb 9247 . . . 4 (𝐴 (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴))
6 ranksnb 9247 . . . 4 (𝐵 (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
7 uneq12 4119 . . . 4 (((rank‘{𝐴}) = suc (rank‘𝐴) ∧ (rank‘{𝐵}) = suc (rank‘𝐵)) → ((rank‘{𝐴}) ∪ (rank‘{𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
85, 6, 7syl2an 598 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
94, 8eqtrd 2859 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐵})) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
10 df-pr 4552 . . 3 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
1110fveq2i 6661 . 2 (rank‘{𝐴, 𝐵}) = (rank‘({𝐴} ∪ {𝐵}))
12 rankon 9215 . . . 4 (rank‘𝐴) ∈ On
1312onordi 6282 . . 3 Ord (rank‘𝐴)
14 rankon 9215 . . . 4 (rank‘𝐵) ∈ On
1514onordi 6282 . . 3 Ord (rank‘𝐵)
16 ordsucun 7530 . . 3 ((Ord (rank‘𝐴) ∧ Ord (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘𝐵)) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵)))
1713, 15, 16mp2an 691 . 2 suc ((rank‘𝐴) ∪ (rank‘𝐵)) = (suc (rank‘𝐴) ∪ suc (rank‘𝐵))
189, 11, 173eqtr4g 2884 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 399   = wceq 1538  wcel 2115  cun 3917  {csn 4549  {cpr 4551   cuni 4824  cima 5545  Ord word 6177  Oncon0 6178  suc csuc 6180  cfv 6343  𝑅1cr1 9182  rankcrnk 9183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7571  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-r1 9184  df-rank 9185
This theorem is referenced by:  rankopb  9272  rankpr  9277  r1limwun  10150  rankaltopb  33465
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