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Theorem rankopb 9127
 Description: The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
rankopb ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankopb
StepHypRef Expression
1 dfopg 4708 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21fveq2d 6542 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = (rank‘{{𝐴}, {𝐴, 𝐵}}))
3 snwf 9084 . . 3 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
4 prwf 9086 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → {𝐴, 𝐵} ∈ (𝑅1 “ On))
5 rankprb 9126 . . 3 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, 𝐵} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, 𝐵}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
63, 4, 5syl2an2r 681 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, 𝐵}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
7 snsspr1 4654 . . . . . 6 {𝐴} ⊆ {𝐴, 𝐵}
8 ssequn1 4077 . . . . . 6 ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵})
97, 8mpbi 231 . . . . 5 ({𝐴} ∪ {𝐴, 𝐵}) = {𝐴, 𝐵}
109fveq2i 6541 . . . 4 (rank‘({𝐴} ∪ {𝐴, 𝐵})) = (rank‘{𝐴, 𝐵})
11 rankunb 9125 . . . . 5 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, 𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, 𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
123, 4, 11syl2an2r 681 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, 𝐵})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})))
13 rankprb 9126 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
1410, 12, 133eqtr3a 2855 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))
15 suceq 6131 . . 3 (((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
1614, 15syl 17 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, 𝐵})) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
172, 6, 163eqtrd 2835 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081   ∪ cun 3857   ⊆ wss 3859  {csn 4472  {cpr 4474  ⟨cop 4478  ∪ cuni 4745   “ cima 5446  Oncon0 6066  suc csuc 6068  ‘cfv 6225  𝑅1cr1 9037  rankcrnk 9038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-r1 9039  df-rank 9040 This theorem is referenced by:  rankop  9133
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