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Theorem rankaltopb 32384
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 8837 . . 3 (𝐵 (𝑅1 “ On) → {𝐵} ∈ (𝑅1 “ On))
2 df-altop 32363 . . . . . 6 𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
32fveq2i 6347 . . . . 5 (rank‘⟪𝐴, 𝐵⟫) = (rank‘{{𝐴}, {𝐴, {𝐵}}})
4 snwf 8837 . . . . . . 7 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
54adantr 472 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → {𝐴} ∈ (𝑅1 “ On))
6 prwf 8839 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → {𝐴, {𝐵}} ∈ (𝑅1 “ On))
7 rankprb 8879 . . . . . 6 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
85, 6, 7syl2anc 696 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
93, 8syl5eq 2798 . . . 4 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
10 snsspr1 4482 . . . . . . . 8 {𝐴} ⊆ {𝐴, {𝐵}}
11 ssequn1 3918 . . . . . . . 8 ({𝐴} ⊆ {𝐴, {𝐵}} ↔ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}})
1210, 11mpbi 220 . . . . . . 7 ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}}
1312fveq2i 6347 . . . . . 6 (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = (rank‘{𝐴, {𝐵}})
14 rankunb 8878 . . . . . . 7 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
155, 6, 14syl2anc 696 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
16 rankprb 8879 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘{𝐴, {𝐵}}) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
1713, 15, 163eqtr3a 2810 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
18 suceq 5943 . . . . 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 2786 . . 3 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
211, 20sylan2 492 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
22 ranksnb 8855 . . . . 5 (𝐵 (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
2322uneq2d 3902 . . . 4 (𝐵 (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)))
24 suceq 5943 . . . 4 (((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
25 suceq 5943 . . . 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 473 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
2821, 27eqtrd 2786 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 383   = wceq 1624  wcel 2131  cun 3705  wss 3707  {csn 4313  {cpr 4315   cuni 4580  cima 5261  Oncon0 5876  suc csuc 5878  cfv 6041  𝑅1cr1 8790  rankcrnk 8791  caltop 32361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-r1 8792  df-rank 8793  df-altop 32363
This theorem is referenced by: (None)
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