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Theorem rankaltopb 32211
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 8710 . . 3 (𝐵 (𝑅1 “ On) → {𝐵} ∈ (𝑅1 “ On))
2 df-altop 32190 . . . . . 6 𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}}
32fveq2i 6232 . . . . 5 (rank‘⟪𝐴, 𝐵⟫) = (rank‘{{𝐴}, {𝐴, {𝐵}}})
4 snwf 8710 . . . . . . 7 (𝐴 (𝑅1 “ On) → {𝐴} ∈ (𝑅1 “ On))
54adantr 480 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → {𝐴} ∈ (𝑅1 “ On))
6 prwf 8712 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → {𝐴, {𝐵}} ∈ (𝑅1 “ On))
7 rankprb 8752 . . . . . 6 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
85, 6, 7syl2anc 694 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘{{𝐴}, {𝐴, {𝐵}}}) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
93, 8syl5eq 2697 . . . 4 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
10 snsspr1 4377 . . . . . . . 8 {𝐴} ⊆ {𝐴, {𝐵}}
11 ssequn1 3816 . . . . . . . 8 ({𝐴} ⊆ {𝐴, {𝐵}} ↔ ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}})
1210, 11mpbi 220 . . . . . . 7 ({𝐴} ∪ {𝐴, {𝐵}}) = {𝐴, {𝐵}}
1312fveq2i 6232 . . . . . 6 (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = (rank‘{𝐴, {𝐵}})
14 rankunb 8751 . . . . . . 7 (({𝐴} ∈ (𝑅1 “ On) ∧ {𝐴, {𝐵}} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
155, 6, 14syl2anc 694 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘({𝐴} ∪ {𝐴, {𝐵}})) = ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})))
16 rankprb 8752 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘{𝐴, {𝐵}}) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
1713, 15, 163eqtr3a 2709 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → ((rank‘{𝐴}) ∪ (rank‘{𝐴, {𝐵}})) = suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
18 suceq 5828 . . . . 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 2685 . . 3 ((𝐴 (𝑅1 “ On) ∧ {𝐵} ∈ (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
211, 20sylan2 490 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟪𝐴, 𝐵⟫) = suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})))
22 ranksnb 8728 . . . . 5 (𝐵 (𝑅1 “ On) → (rank‘{𝐵}) = suc (rank‘𝐵))
2322uneq2d 3800 . . . 4 (𝐵 (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)))
24 suceq 5828 . . . 4 (((rank‘𝐴) ∪ (rank‘{𝐵})) = ((rank‘𝐴) ∪ suc (rank‘𝐵)) → suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
25 suceq 5828 . . . 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 481 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → suc suc ((rank‘𝐴) ∪ (rank‘{𝐵})) = suc suc ((rank‘𝐴) ∪ suc (rank‘𝐵)))
2821, 27eqtrd 2685 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 1523  wcel 2030  cun 3605  wss 3607  {csn 4210  {cpr 4212   cuni 4468  cima 5146  Oncon0 5761  suc csuc 5763  cfv 5926  𝑅1cr1 8663  rankcrnk 8664  caltop 32188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666  df-altop 32190
This theorem is referenced by: (None)
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