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Theorem rankaltopb 31720
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 8617 . . 3 (𝐵 (𝑅1 “ On) → {𝐵} ∈ (𝑅1 “ On))
2 df-altop 31699 . . . . . 6 𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
32fveq2i 6153 . . . . 5 (rank‘⟪𝐴, 𝐵⟫) = (rank‘{{𝐴}, {𝐴, {𝐵}}})
4 snwf 8617 . . . . . . 7 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
54adantr 481 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → {𝐴} ∈ (𝑅1 “ On))
6 prwf 8619 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → {𝐴, {𝐵}} ∈ (𝑅1 “ On))
7 rankprb 8659 . . . . . 6 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
85, 6, 7syl2anc 692 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
93, 8syl5eq 2672 . . . 4 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
10 snsspr1 4318 . . . . . . . 8 {𝐴} ⊆ {𝐴, {𝐵}}
11 ssequn1 3766 . . . . . . . 8 ({𝐴} ⊆ {𝐴, {𝐵}} ↔ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}})
1210, 11mpbi 220 . . . . . . 7 ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}}
1312fveq2i 6153 . . . . . 6 (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = (rank‘{𝐴, {𝐵}})
14 rankunb 8658 . . . . . . 7 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
155, 6, 14syl2anc 692 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
16 rankprb 8659 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘{𝐴, {𝐵}}) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
1713, 15, 163eqtr3a 2684 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
18 suceq 5752 . . . . 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 2660 . . 3 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
211, 20sylan2 491 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
22 ranksnb 8635 . . . . 5 (𝐵 (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
2322uneq2d 3750 . . . 4 (𝐵 (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)))
24 suceq 5752 . . . 4 (((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
25 suceq 5752 . . . 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 2660 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 384   = wceq 1480  wcel 1992  cun 3558  wss 3560  {csn 4153  {cpr 4155   cuni 4407  cima 5082  Oncon0 5685  suc csuc 5687  cfv 5850  𝑅1cr1 8570  rankcrnk 8571  caltop 31697
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-r1 8572  df-rank 8573  df-altop 31699
This theorem is referenced by: (None)
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