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Theorem rankaltopb 34502
Description: Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rankaltopb ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))

Proof of Theorem rankaltopb
StepHypRef Expression
1 snwf 9703 . . 3 (𝐵 (𝑅1 “ On) → {𝐵} ∈ (𝑅1 “ On))
2 df-altop 34481 . . . . . 6 𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
32fveq2i 6842 . . . . 5 (rank‘⟪𝐴, 𝐵⟫) = (rank‘{{𝐴}, {𝐴, {𝐵}}})
4 snwf 9703 . . . . . . 7 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
54adantr 481 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → {𝐴} ∈ (𝑅1 “ On))
6 prwf 9705 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → {𝐴, {𝐵}} ∈ (𝑅1 “ On))
7 rankprb 9745 . . . . . 6 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
85, 6, 7syl2anc 584 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
93, 8eqtrid 2789 . . . 4 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
10 snsspr1 4772 . . . . . . . 8 {𝐴} ⊆ {𝐴, {𝐵}}
11 ssequn1 4138 . . . . . . . 8 ({𝐴} ⊆ {𝐴, {𝐵}} ↔ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}})
1210, 11mpbi 229 . . . . . . 7 ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}}
1312fveq2i 6842 . . . . . 6 (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = (rank‘{𝐴, {𝐵}})
14 rankunb 9744 . . . . . . 7 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
155, 6, 14syl2anc 584 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
16 rankprb 9745 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘{𝐴, {𝐵}}) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
1713, 15, 163eqtr3a 2801 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
18 suceq 6381 . . . . 5 (((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
1917, 18syl 17 . . . 4 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
209, 19eqtrd 2777 . . 3 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
211, 20sylan2 593 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
22 ranksnb 9721 . . . . 5 (𝐵 (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
2322uneq2d 4121 . . . 4 (𝐵 (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)))
24 suceq 6381 . . . 4 (((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
25 suceq 6381 . . . 4 (suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
2623, 24, 253syl 18 . . 3 (𝐵 (𝑅1 “ On) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
2726adantl 482 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
2821, 27eqtrd 2777 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cun 3906  wss 3908  {csn 4584  {cpr 4586   cuni 4863  cima 5634  Oncon0 6315  suc csuc 6317  cfv 6493  𝑅1cr1 9656  rankcrnk 9657  caltop 34479
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-om 7795  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-r1 9658  df-rank 9659  df-altop 34481
This theorem is referenced by: (None)
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